Advances in Applied Mathematics and Global Optimization

ADVANCES IN APPLIED MATHEMATICS AND GLOBAL OPTIMIZATION IN HONOR OF GILBERT STRANG Advances in Mechanics and Mathemati...

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ADVANCES IN APPLIED MATHEMATICS AND GLOBAL OPTIMIZATION IN HONOR OF GILBERT STRANG

Advances in Mechanics and Mathematics VOLUME 17 Series Editors David Y. Gao (Virginia Polytechnic Institute and State University) Ray W. Ogden (University of Glasgow)

Advisory Board Ivar Ekeland (University of British Columbia, Vancouver) Tim Healey (Cornell University, USA) Kumbakonam Rajagopal (Texas A&M University, USA) Tudor Ratiu (École Polytechnique Fédérale, Lausanne) David J. Steigmann (University of California, Berkeley)

Aims and Scope Mechanics and mathematics have been complementary partners since Newton’s time, and the history of science shows much evidence of the beneficial influence of these disciplines on each other. The discipline of mechanics, for this series, includes relevant physical and biological phenomena such as: electromagnetic, thermal, quantum effects, biomechanics, nanomechanics, multiscale modeling, dynamical systems, optimization and control, and computational methods. Driven by increasingly elaborate modern technological applications, the symbiotic relationship between mathematics and mechanics is continually growing. The increasingly large number of specialist journals has generated a complementarity gap between the partners, and this gap continues to widen. Advances in Mechanics and Mathematics is a series dedicated to the publication of the latest developments in the interaction between mechanics and mathematics and intends to bridge the gap by providing interdisciplinary publications in the form of monographs, graduate texts, edited volumes, and a special annual book consisting of invited survey articles.

ADVANCES IN APPLIED MATHEMATICS AND GLOBAL OPTIMIZATION IN HONOR OF GILBERT STRANG

Edited By David Y. Gao Virginia Polytechnic Institute, Blacksburg, VA Hanif D. Sherali Virginia Polytechnic Institute, Blacksburg, VA

Editors: David Y. Gao Department Mathematics Virginia Polytechnic Institute Blacksburg, VA 24061 [email protected] Series Editors: David Y. Gao Department of Mathematics Virginia Polytechnic Institute Blacksburg, VA 24061 [email protected]

ISBN 978-0-387-75713-1 DOI 10.1007/978-0-387-75714-8

Hanif D. Sherali Department Mathematics Virginia Polytechnic Institute Blacksburg, VA 24061 [email protected]

Ray W. Ogden Department of Mathematics University of Glasgow Glasgow, Scotland, UK [email protected]

e-ISBN 978-0-387-75714-8

Library of Congress Control Number: 2009921139 Mathematics Subject Classification (2000): 90-00, 49-00, 74-00, 65, 74, 81, 92 ©Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

This book is dedicated to Professor Gilbert Strang on the occasion of his 70th birthday

Gilbert Strang

Gil Strang in his MIT office

Contents

Series Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Constrained Optimism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Gilbert Strang Biographical Summary of Gilbert Strang . . . . . . . . . . . . . . . . . . . . . xvii List of Publications of Gilbert Strang . . . . . . . . . . . . . . . . . . . . . . . . . xix 1

Maximum Flows and Minimum Cuts in the Plane . . . . . . . . Gilbert Strang

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Variational Principles and Residual Bounds for Nonpotential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Giles Auchmuty

3

Adaptive Finite Element Solution of Variational Inequalities with Application in Contact Problems . . . . . . . 25 Viorel Bostan and Weimin Han

4

Time—Frequency Analysis of Brain Neurodynamics . . . . . . 107 W. Art Chaovalitwongse, W. Suharitdamrong, and P.M. Pardalos

5

Nonconvex Optimization for Communication Networks . . . 137 Mung Chiang

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Multilevel (Hierarchical) Optimization: Complexity Issues, Optimality Conditions, Algorithms . . . . . . . . . . . . . . . 197 Altannar Chinchuluun, Panos M. Pardalos, and Hong-Xuan Huang

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Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Antoine Deza, Tam´ as Terlaky, and Yuriy Zinchenko

8

Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization . . . . . . . . . . 257 David Y. Gao and Hanif D. Sherali

9

Quantum Computation and Quantum Operations . . . . . . . . . 327 Stan Gudder

10 Ekeland Duality as a Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Jean-Paul Penot 11 Global Optimization in Practice: State of the Art and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 János D. Pintér 12 Two-Stage Stochastic Mixed-Integer Programs: Algorithms and Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Hanif D. Sherali and Xiaomei Zhu 13 Dualistic Riemannian Manifold Structure Induced from Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Jun Zhang and Hiroshi Matsuzoe 14 NMR Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Zhigang Zhang, Goong Chen, Zijian Diao, and Philip R. Hemmer

Series Preface

As any human activity needs goals, mathematical research needs problems. –David Hilbert Mechanics is the paradise of mathematical sciences. –Leonardo da Vinci

Mechanics and mathematics have been complementary partners since Newton’s time, and the history of science shows much evidence of the beneficial influence of these disciplines on each other. Driven by increasingly elaborate modern technological applications, the symbiotic relationship between mathematics and mechanics is continually growing. However, the increasingly large number of specialist journals has generated a duality gap between the partners, and this gap is growing wider. Advances in Mechanics and Mathematics (AMMA) is intended to bridge the gap by providing multidisciplinary publications that fall into the two following complementary categories: 1. An annual book dedicated to the latest developments in mechanics and mathematics; 2. Monographs, advanced textbooks, handbooks, edited volumes, and selected conference proceedings. The AMMA annual book publishes invited and contributed comprehensive research and survey articles within the broad area of modern mechanics and applied mathematics. The discipline of mechanics, for this series, includes relevant physical and biological phenomena such as: electromagnetic, thermal, and quantum effects, biomechanics, nanomechanics, multiscale modeling, dynamical systems, optimization and control, and computation methods. Especially encouraged are articles on mathematical and computational models and methods based on mechanics and their interactions with other fields. All contributions will be reviewed so as to guarantee the highest possible scientific standards. Each chapter will reflect the most recent achievements in xi

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Series Preface

the area. The coverage should be conceptual, concentrating on the methodological thinking that will allow the nonspecialist reader to understand it. Discussion of possible future research directions in the area is welcome. Thus, the annual volumes will provide a continuous documentation of the most recent developments in these active and important interdisciplinary fields. Chapters published in this series could form bases from which possible AMMA monographs or advanced textbooks could be developed. Volumes published in the second category contain review/research contributions covering various aspects of the topic. Together these will provide an overview of the state-of-the-art in the respective field, extending from an introduction to the subject right up to the frontiers of contemporary research. Certain multidisciplinary topics, such as duality, complementarity, and symmetry in mechanics, mathematics, and physics are of particular interest. The Advances in Mechanics and Mathematics series is directed to all scientists and mathematicians, including advanced students (at the doctoral and postdoctoral levels) at universities and in industry who are interested in mechanics and applied mathematics.

David Y. Gao Ray W. Ogden

Preface

Complementarity and duality are closely related, multidisciplinary topics that pervade all natural phenomena, and form the basis for solving many underlying nonconvex analysis and global optimization problems that arise in science and engineering. During the last forty years, much research has been devoted to the development of mathematical modeling, theory, and computational methods in this area. The field has now matured in convex systems, especially in linear programming, engineering mechanics and design, mathematical physics, economics, optimization, and control. However, in nonconvex systems many fundamental problems still remain unsolved. In view of the importance of complementarity—duality theory and methods in applied mathematics and mathematical programming, and in order to bridge the ever-increasing gap between global optimization and engineering science, the First International Conference on Complementarity, Duality, and Global Optimization (CDGO) was held at Virginia Tech, Blacksburg, August 15—17, 2005, under the sponsorship of the National Science Foundation. This conference brought together more than 100 world-class researchers from interdisciplinary fields of industrial engineering, operations research, pure and applied math, engineering mechanics, electrical engineering, psychology, management science, civil engineering, and computational science. This conference spawned some new trends in optimization and engineering science, and has stimulated young faculty and students to venture into this rich domain of research. This AMMA Annual contains eleven chapters from selected lectures presented at the First International Conference on Complementarity, Duality, and Global Optimization (CDGO) in August 2005 and three invited chapters by experts in computational mathematics and quantum computations. These chapters deal with fundamental theory, methods, and applications of complementarity, duality, and global optimization in multidisciplinary fields of global optimization, nonconvex mechanics, and computational science, as well as the very contemporary topic of quantum computing, which is at the forefront of

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the scientific and technological research and development of the twenty-first century. This special volume is dedicated to Gilbert Strang on the occasion of his 70th birthday. Professor Strang is a world-renowned mathematician not only by his scientific contributions, but also by his personal character which exemplifies what a real scientist should possess. During his exceptional academic career and social activities spanning almost a half-century, Dr. Strang has had a profound influence on the development of interdisciplinary fields in applied mathematics, mechanics, and engineering science, including the field of complementary duality in calculus of variations, optimization, numerical methods, and mathematical education. The unified beauty of duality can be viewed throughout his celebrated textbooks, lecture notes, essays, and scientific publications, which will continue to influence several generations in the broad field of mathematical sciences. Credit for this special volume is to be shared by all the eminent contributing authors. As editors, we are deeply indebted to them. Our special thanks also go to Ann Kostant and her team, and especially to Elizabeth Loew at Springer, for all their great enthusiasm and professional help in expediting the publication of this annual volume.

May 2008 Blacksburg, VA

David Y. Gao Hanif D. Sherali

Constrained Optimism Gilbert Strang

The editors have kindly invited me to write a few words of introduction to this volume. They even expressed the hope that I would go beyond mathematics, to say something about my own life experiences. I think every reader will recognize how hard it is (meaning impossible) to do that properly. If I choose a single word to describe an approach to the complications of life (and of mathematics too), it would be “optimism.” Eventually I realized that, if you allow that word in its mathematical sense too, this whole book is for optimists. If I may give one instance of my own optimism, it has come from writing textbooks. I enjoy the hopeless effort to express simple ideas clearly. Beyond that, I have come to expect (without knowing any reason, perhaps this defines an optimist) that the connections between all the pieces of the book will somehow appear. Suddenly a topic fits into its right place. This irrational certainty may also be the experience of a hopeful novelist who doesn’t know how the characters will interact and how the plot will turn out. Looking seriously at this approach, to applied mathematics or to life, an unconstrained optimism is hard to justify. Mathematically, an immediate constraint on all of us is that we are “not Gauss.” Far wiser to accept constraints, and continue to optimize. The connection that did finally bring order to my own thinking and writing about applied mathematics and computational engineering was constrained optimization. I now call that the “Fundamental Problem of Scientific Computing.” Examples are everywhere, or those words would not be justified. So many problems involve three steps, and flows in networks are a good model. The potentials at the nodes, and the currents on the edges, are the unknowns (somehow dual). A first step goes from potentials to potential differences (by an edge-node matrix A). The second step relates potential differences to flows (by a matrix C). Ohm’s law is typical, or Hooke’s law, or any constitutive law: linear at first but not forever. The third step is the essential constraint of conservation or continuity or balance of forces, as in Kirchhoff’s current law. This involves the transpose matrix A0.

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The dual role of A and A0 is at first a miracle. A reason begins to emerge through minimization and Lagrange multipliers. If we minimize a quadratic energy with a linear constraint A0 w = f , the optimality conditions lead to a saddle point matrix (“KKT matrix”): ∙ −1 ¸ ∙ ¸ ∙ ¸ C A w b Optimization with constraint: = . A0 0 u f One way to solve this fundamental problem is to eliminate w. The three matrices combine into A0 CA, symmetric and positive definite in the best case. This is the stiffness matrix of the finite element method, or the Laplacian matrix of finite differences and graph theory. It appears everywhere and we don’t know the best way to solve the equation. As a differential equation it is in divergence form with A0 CA = div(c grad). When C is piecewise linear we have mathematical programming, where the primal—dual method has come to the front. The real problems of mechanics and biology (and life) are not linear at all. But remarkably often they still have this form with A0 C (Au). May I thank the editors and authors and readers of the present book. I hope you will accept constraints as inevitable, and go forward.

Biographical Summary of Gilbert Strang

Education 1. 1952—1955 William Barton Rogers Scholar, M.I.T. (S.B. 1955) 2. 1955—1957 Rhodes Scholar, Oxford University (B.A., M.A. 1957) 3. 1957—1959 NSF Fellow, UCLA (Ph.D. 1959)

Positions Held 1. 2. 3. 4. 5.

1959—1961 C.L.E. Moore Instructor, M.I.T. 1961—1962 NATO Postdoctoral Fellow, Oxford University 1962—1964 Assistant Professor of Mathematics, M.I.T. 1964—1970 Associate Professor of Mathematics, M.I.T. 1970— Professor of Mathematics, M.I.T.

Awards and Duties 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Alfred P. Sloan Fellow (1966—1967) Chairman, M.I.T. Committee on Pure Mathematics (1975—1979) Chauvenet Prize, Mathematical Association of America (1976) Council, Society for Industrial and Applied Mathematics (1977—1982) NSF Advisory Panel on Mathematics (1977—1980) (Chairman 1979—1980) CUPM Subcommittee on Calculus (1979—1981) Fairchild Scholar, California Institute of Technology (1980—1981) Honorary Professor, Xian Jiaotong University, China (1980) American Academy of Arts and Sciences (1985) Taft Lecturer, University of Cincinnati (1977) Gergen Lecturer, Duke University (1983) Lonseth Lecturer, Oregon State University (1987) Magnus Lecturer, Colorado State University (2000) Blumberg Lecturer, University of Texas (2001) AMS-SIAM Committee on Applied Mathematics (1990—1992) Vice President for Education, SIAM (1991—1996) MAA Science Policy Committee (1992—1995) xvii

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Biographical Summary of Gilbert Strang

18. Committee on the Undergraduate Program in Mathematics, MAA (1993— 1996) 19. President of SIAM (1999—2000) 20. Chair, Joint Policy Board for Mathematics (1999) 21. Chair, SIAM Committee on Science Policy (2001—2002) 22. Honorary Fellow, Balliol College, Oxford (1999) 23. Honorary Member, Irish Mathematical Society (2002) 24. US National Committee on Mathematics (2001—2004, Chair 2003—2004) 25. Award for Distinguished Service to the Profession, SIAM (2003) 26. Graduate School Teaching Award, MIT (2003) 27. Abel Prize Committee, Oslo (2003—2005) 28. Von Neumann Prize Medal, US Association for Computational Mechanics (2005) 29. Ford Prize for “Pascal Matrices” with Alan Edelman, Mathematical Association of America (2005) 30. Distinguished University Teacher of Mathematics, New England Section, Mathematical Association of America (2006) 31. Franklin and Deborah Tepper Haimo Prize, MAA (2006) 32. Su Buchin Prize, International Congress of Industrial and Applied Mathematics (ICIAM, Zurich, 2007). 33. Henrici Prize, (ICIAM, Zurich, 2007).

Journal Editor 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Numerische Mathematik (Honorary Editor from 1996) International Journal for Numerical Methods in Engineering Archive for Rational Mechanics and Analysis (to 1990) Studies in Applied Mathematics Computer Methods in Applied Mechanics and Engineering (to 2004) SIAM Journal on Numerical Analysis (to 1977) Numerical Functional Analysis and Optimization Physica D: Nonlinear Phenomena (to 1986) Communications in Numerical Methods in Engineering SIAM Journal on Matrix Analysis and Applications (to 1993) Acta Applicandae Mathematicae Proceedings of the Edinburgh Mathematical Society (to 1993) Mathematical Modelling and Numerical Analysis Japan Journal of Applied Mathematics Structural Optimization Royal Society of Edinburgh, Proceedings A Computational Optimization and Applications SIAM Review (to 2001) COSMOS, National University of Singapore Numerical Algorithms (to 2004)

List of Publications of Gilbert Strang

Books (with Russian, Japanese, and Chinese translations of the first two) 1. An Analysis of the Finite Element Method, with George Fix, PrenticeHall (1973). Second Edition: Wellesley-Cambridge Press (2008). 2. Linear Algebra and Its Applications, Academic Press (1976). Second Edition: Harcourt Brace Jovanovich (1980). Third Edition: Brooks/Cole (1988). Fourth Edition: Brooks/Cole (2006). 3. Introduction to Applied Mathematics, Wellesley-Cambridge Press (1986). 4. Nonlinear Partial Differential Equations in Applied Science, H. Fujita, P. Lax, G. Strang, editors, Lecture Notes in Numerical and Applied Analysis 5, Kinokuniya/North Holland (1982). 5. Topics in Nonsmooth Mechanics, J.J. Moreau, P.D. Panagiotopoulos, G. Strang, editors, Birkh¨ auser (1988). 6. Calculus, Wellesley-Cambridge Press (1991). 7. Introduction to Linear Algebra, Wellesley-Cambridge Press (1993). Second Edition (1998). Third Edition (2003). 8. Wavelets and Filter Banks, with Truong Nguyen, Wellesley-Cambridge Press (1996). 9. Linear Algebra, Geodesy, and GPS, with Kai Borre, Wellesley-Cambridge Press (1997). 10. Computational Science and Engineering, Wellesley-Cambridge Press (2007).

Papers in Journals and Books 1. An improvement on the Holzer table based on a suggestion of Rayleigh’s, with S.H. Crandall, J. Appl. Mechanics, Paper 56-A27 (1957). 2. On the order of convergence of the Crank-Nicolson procedure, Journal of Mathematics and Physics 38 (1959) 141—144. 3. Difference methods for mixed boundary-value problems, Duke Math. J. 27 (1960) 221—232. 4. On the Kantorovich inequality, Proc. Amer. Math. Soc. 11 (1960) 468. xix

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5. A note on the joint spectral radius, with G.-C. Rota, Proc. Netherlands Acad. 22 (1960) 379—381. 6. Finite difference techniques for a boundary problem, with L. Ehrlich, J. Riley, and B.A. Troesch, J. Soc. Ind. Appl. Math. (1961). 7. Eigenvalues of Jordan products, Amer. Math. Monthly 69 (1962) 37—40. 8. Trigonometric polynomials and difference methods of maximum accuracy, Journal of Mathematics and Physics 41 (1962) 147—154. 9. Polynomial approximation of Bernstein type, Trans. Amer. Math. Soc. 105 (1962) 525—535. 10. Comparison theorems for supremum norms, with H. Schneider, Numerische Math. 4 (1962) 15—20. 11. Accurate partial difference methods I: Linear Cauchy problems, Arch. Rat. Mech. Anal. 12 (1963). 12. Accurate partial difference methods II: Non-linear problems, Numerische Math. 6 (1964) 37—46. 13. Wiener-Hopf difference equations, J. Math. Mechanics 13 (1964) 85—96. 14. Unbalanced polynomials and difference methods for mixed problems, SIAM J. Numer. Anal. 2 (1964) 46—51. 15. Necessary and insufficient conditions for well-posed Cauchy problems, J. Diff. Eq. 2 (1966) 107—114. 16. Matrix theorems for partial differential and difference equations, with J. Miller, Math. Scand. 18 (1966) 113—133. 17. Implicit difference methods for initial-boundary value problems, J. Math. Anal. Appl. 16 (1966) 188—198. 18. On strong hyperbolicity, J. Math. Kyoto Univ. 6 (1967) 397—417. 19. A variant of Caratheodory’s problem, Proc. Edinburgh Math. Soc. 16 (1968) 43—48. 20. The nucleus of a set, Canad. Math. Bull. 11 (1968) 65—72. 21. On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968) 506—517. 22. Approximating semigroups and the consistency of difference schemes, Proc. Amer. Math. Soc. 20 (1969) 1—7. 23. Hyperbolic initial-boundary value problems in two unknowns, J. Diff. Eq. 6 (1969) 161—171. 24. On numerical ranges and holomorphic semigroups, J. d’Analyse Math. 22 (1969) 299—318. 25. On multiple characteristics and the Levi-Lax conditions for hyperbolicity, Arch. Rat. Mech. Anal. 33 (1969) 358—373. 26. Fourier analysis of the finite element method in Ritz-Galerkin theory, with G. Fix, Stud. Appl. Math. 48 (1969) 265—273. 27. Toeplitz operators in a quarter-plane, Bull. Amer. Math. Soc. 76 (1970) 1303—1307. 28. The correctness of the Cauchy problem, with H. Flaschka, Adv. Math. 6 (1971) 347—349.


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29. The finite element method and approximation theory, SYNSPADE Proceedings, Academic Press (1971) 547—584. 30. The change in solution due to change in domain, with A. Berger, AMS Symposium on Partial Differential Equations, Berkeley (1971) 199—206. 31. Approximation in the finite element method, Numerische Math. 19 (1972) 81—98. 32. Approximate boundary conditions in the finite element method, with R. Scott and A. Berger, Symposia Mathematica X, Istituto Nationale di Alta Matematica (1972) 295—313. 33. Variational crimes in the finite element method, The Mathematical Foundations of the Finite Element Method, ed. by A.K. Aziz, Academic Press (1973) 689—710. 34. A Fourier analysis of the finite element variational method, with G. Fix, Constructive Aspects of Functional Analysis, Edizioni Cremonese, Rome (1973) 795—840. 35. Piecewise polynomials and the finite element method, AMS Bulletin 79 (1973) 1128—1137. 36. Optimal conditioning of matrices, with C. McCarthy, SIAM J. Numer. Anal. 10 (1973) 370—388. 37. The dimension of piecewise polynomial spaces and one-sided approximation, Proc. Conference on Numerical Analysis, Dundee, Springer Lecture Notes 363 (1974) 144—152. 38. One-Sided Approximation and Plate Bending, Lecture Notes in Computer Science 11, Springer-Verlag (1974) 140—155. 39. One-sided approximation and variational inequalities, with U. Mosco, Bull. Amer. Math. Soc. 80 (1974) 308—312. 40. The finite element method–linear and nonlinear applications, Proc. Inter. Congress of Mathematicians, Vancouver (1974). 41. Free boundaries and finite elements in one dimension, with W. Hager, Math. Comp. 29 (1975) 1020—1031. 42. A homework exercise in finite elements, Int. J. Numer. Meth. Eng. 11 (1977) 411—418. 43. Some recent contributions to plasticity theory, J. Franklin Institute 302 (1977) 429—442. 44. Discrete plasticity and the complementarity problem, Proceedings U.S.Germany Symposium: Formulations and Computational Algorithms in Finite Element Analysis, M.I.T. Press (1977) 839—854. 45. Uniqueness in the theory of variational inequalities, Adv. Math. 22 (1976) 356—363. 46. A minimax problem in plasticity theory, Functional Analysis Methods in Numerical Analysis, ed. M.Z. Nashed, Springer Lecture Notes 701, Springer (1979) 319—333. 47. A family of model problems in plasticity, Proc. Symp. Computing Methods in Applied Sciences, ed. R. Glowinski and J.L. Lions, Springer Lecture Notes 704, Springer (1979) 292—308.

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48. The saddle point of a differential program, with H. Matthies and E. Christiansen, Energy Methods in Finite Element Analysis, ed. by R. Glowinski, E. Rodin, and O.C. Zienkiewicz, John Wiley (1979). 49. The solution of nonlinear finite element equations, with H. Matthies, Int. J. Numer. Meth. Eng. 14 (1979) 1613—1626. 50. Mathematical and computational methods in plasticity, with H. Matthies and R. Temam, Proc. IUTAM Symp. on Variational Methods in the Mechanics of Solids, S. Nemat-Nasser, ed., Pergamon (1980) 20—28. 51. Spectral decomposition in advection-diffusion analysis by finite element methods, with R. Nickell and D. Gartling, Proc. FENOMECH Symp., Stuttgart (1978); Comput. Meth. Appl. Mech. Eng. 17 (1979) 561—580. 52. Existence de solutions relaxes pour les equations de la plasticite, with R. Temam, Comptes Rendus Acad. Sc. Paris 287 (1978) 515—519. 53. Functions of bounded deformation, with R. Temam, Arch. Rat. Mech. Anal. 75 (1980) 7—21. 54. Numerical computations in nonlinear mechanics, with H. Matthies, Paper 79-PVP-103, Amer. Soc. Mech. Eng. (1979); Proceedings of the 4th Symposium on Computing Methods in Applied Sciences and Engineering, ed. R. Glowinski and J.L. Lions, 517—525, North-Holland (1980). 55. Duality and relaxation in the variational problems of plasticity, with R. Temam, J. Mecanique 19 (1980) 1—35. 56. The quasi-Newton method in finite element calculations, Chapter 20 in Computational Methods in Nonlinear Mechanics, J.T. Oden, ed., NorthHolland (1980). 57. The application of quasi-Newton methods in fluid mechanics, with M. Engelman and K.J. Bathe, Int. J. Numer. Meth. Eng. 17 (1981) 707— 718. 58. A problem in capillarity and plasticity, with R. Temam, Nondifferentiable and Variational Techniques in Optimization, D.C. Sorenson, R.J.B. Wets, eds., Mathematical Programming Study 17 (1982) 91—102. 59. Optimal design for torsional rigidity, with R. Kohn, Proc. Int. Symp. on Mixed and Hybrid Finite Element Methods, Atlanta (1981). 60. Optimal design of cylinders in shear, with R. Kohn, MAFELAP Conference, Brunel (1981). 61. The width of a chair, Amer. Math. Monthly 89 (1982) 529—534. 62. Structural design optimization, homogenization, and relaxation of variational problems, with R. Kohn, Proceedings of Conference on Disordered Media, Lecture Notes in Physics 154, Springer-Verlag (1982). 63. Hencky—Prandtl nets and constrained Michell trusses, with R. Kohn, Conference on Optimum Structural Design, Tucson (1981), Comput. Meth. Appl. Mech. Eng. 36 (1983) 207—222. 64. The optimal accuracy of difference schemes, with A. Iserles, Trans. Amer. Math. Soc. 277 (1983) 770—803. 65. Duality in the classroom, Amer. Math. Monthly 91 (1984) 250—254. 66. Maximal flow through a domain, Math. Program. 26 (1983) 123—143.


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67. Barriers to stability, with A. Iserles, SIAM J. Numer. Anal. 20 (1983) 1251—1257. 68. L1 and L∞ approximation of vector fields in the plane, Nonlinear Partial Differential Equations in Applied Science, H. Fujita, P. Lax, and G. Strang, eds., Lecture Notes in Num. Appl. Anal. 5 (1982) 273—288. 69. Notes on softening and local instability, with M. Abdel-Naby, in Computational Aspects of Penetration Mechanics, Springer Lecture Notes in Engineering 3, J. Chandra and J. Flaherty, eds. (1983). 70. A negative results for nonnegative matrices, J. Xian Jiaotong Univ. 17 (1983) 69—72. 71. Numerical and biological shape optimization, with A. Philpott, in Unification of Finite Element Methods, Math. Studies 94, H. Kardestuncer, ed., North-Holland (1984). 72. Explicit relaxation of a variational problem in optimal design, with R. Kohn, Bull. Amer. Math. Soc. 9 (1983) 211—214. 73. Optimal design and relaxation of variational problems, with R. Kohn, Commun. Pure Appl. Math. 39 (1986) 113—137 (Part I), 139—182 (Part II), 353—377 (Part III). 74. The constrained least gradient problem, with R. Kohn, in Non-Classical Continuum Mechanics, R. Knops and A. Lacey, eds., Cambridge University Press (1987). 75. The optimal design of a two-way conductor, with R. Kohn, in Nonsmooth Mechanics, P.D. Panagiotopoulos et al, eds., Birkh¨ auser (1987). 76. Fibered structures in optimal design, with R. Kohn, Ordinary and Partial Differential Equations, B. Sleeman and R. Jarvis, eds., Pitman Research Notes 157, Longman (1987). 77. Optimal design in elasticity and plasticity, with R. Kohn, Int. J. Numer. Meth. Eng. 22 (1986) 183—188. 78. A framework for equilibrium equations, SIAM Rev. 30 (1988) 283—297. 79. Karmarkar’s algorithm in a nutshell, SIAM News 18 (1985) 13. 80. Karmarkar’s algorithm and its place in applied mathematics, Math. Intelligencer 9 (1987) 4—10. 81. A proposal for Toeplitz matrix calculations, Stud. Appl. Math. 74 (1986) 171—176. 82. The Toeplitz-circulant eigenvalue problem, with A. Edelman, pp. 109— 117 in Oakland Conf. on PDE’s and Applied Mathematics, L. Bragg and J. Dettman, eds., Longman (1987). 83. Patterns in linear algebra, Amer. Math. Monthly 96 (1989) 105—117. 84. Paradox lost: Natural boundary conditions in the Ritz- Galerkin method, with J. Storch, Int. J. Numer. Meth. Eng. 26 (1988) 2255—2266. 85. Dual extremum principles in finite elastoplastic deformation, with Y. Gao, Acta Appl. Math. 17 (1989) 257—268. 86. Toeplitz equations by conjugate gradients with circulant preconditioner, with R. Chan, SIAM J. Sci. Stat. Comp. 10 (1989) 104—119.

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Chapter 1

Maximum Flows and Minimum Cuts in the Plane Gilbert Strang

Summary. A continuous maximum flow problem finds the largest t such that div v = t F (x, y) is possible with a capacity constraint k(v1 , v2 )k ≤ c(x, y). The dual problem finds a minimum cut ∂S which is filled to capacity by the flow through it. This model problem has found increasing application in medical imaging, and the theory continues to develop (along with new algorithms). Remaining difficulties include explicit streamlines for the maximum flow, and constraints that are analogous to a directed graph. Key words: Maximum flow, minimum cut, capacity constraint, Cheeger

1.1 Introduction This chapter returns to a special class of problems (partial differential equations with inequality constraints) in continuous linear programming. They describe flow through a domain Ω, in analogy with flow along the edges of a graph. The flow is maximized subject to a capacity constraint. The key to the solution is the dual problem, which looks for a set S ⊂ Ω from which no more flow is possible. The boundary of S is the minimum cut, and it is filled to capacity by the maximum flow. In the discrete case, Kirchhoff’s current law that “flow in = flow out” must hold at every interior node of the network. The maximum flow is the largest flow from source to sink, subject to Kirchhoff’s equation at the nodes and capacity constraints on the edges. This fits the standard framework of linear programming, and Kirchhoff’s incidence matrix (of 1s, −1s, and 0s) has remarkable properties that lead to an attractive theory. Our purpose is Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A., e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_1, © Springer Science+Business Media, LLC 2009

1

2

Gilbert Strang

to point to a maximum flow—minimum cut theorem in the continuous case, and to introduce new questions. The principal unknown is the vector v(x, y) that gives the magnitude and direction of the flow. On a plane domain this is v = (v1 (x, y), v2 (x, y)). The analogue of Kirchhoff’s matrix is the divergence operator: Conservation:

div v =

∂v1 ∂v2 + = tF (x, y) in Ω. ∂x ∂y

(1.1)

That source/sink term tF (x, y) might be zero or nonzero in the interior of the flow domain Ω. There may also be a source term tf (x, y) on the boundary ∂Ω, in closer analogy with the discrete source and sink in classical network flow. With n as the unit normal vector to ∂Ω, the (possible) boundary sources and sinks are given by a Neumann condition: Boundary sources: v · n = tf (x, y) on ∂ΩN .

(1.2)

Our examples involve F but not f . We only note the case ∂ΩN = ∂Ω, when f is prescribed on the whole boundary. Then the divergence theorem RR R div v dx dy = v · n ds imposes a compatibility condition on F and f : ZZ Z Compatibility: F (x, y) dx dy = f (x, y) ds if ∂ΩN = ∂Ω. (1.3) Ω

∂Ω

Now comes the key inequality, a limit on the flow. The vector field v(x, y) is subject to a capacity constraint, which makes the problem nonlinear. In our original paper [31] this constraint measured (v1 , v2 ) always in the 2 norm at each point: q (1.4) Capacity: |v(x, y)| = v12 + v22 ≤ c(x, y) in Ω. A more general condition would require v(x, y) to lie in a convex set K(x, y): v(x, y) ∈ K(x, y) for all x, y in Ω.

(1.5)

A typical maximal flow problem in the domain Ω is Maximize t subject to (1.1), (1.2), and (1.4). In returning to this maximal flow problem, our goal is to highlight four questions that were not originally considered. Fortunately there has been good progress by several authors, and partial answers are available. But the new tools are not yet all-powerful, as we illustrate with a challenge problem (uniform source F = 1 and capacity c = 1 with Ω = unit square). This continues to resist explicit solution for the velocity vector v: Challenge: Maximize t so that div v = t with |v| ≤ 1 in Ω.

(1.6)

1 Maximum Flows and Minimum Cuts in the Plane

3

The intriguing aspect of this problem is that we can identify the minimal cut. √ Therefore we know the maximal flow factor t = 2 + π, from the capacity across that cut ∂S. Determining ∂S is a constrained isoperimetric problem that is pleasant to solve (and raises new questions). What we do not know is the flow vector v inside the square ! Optimality tells us the magnitude and direction of v only along the cut, described below. We apologize for the multiplication of new challenges, when the proper goal of a chapter should be new solutions.

1.2 New Questions and Applications The continuous maximal flow problem is attracting a small surge of interest. We mention recent papers that carry the problem forward in several directions:

1. Grieser [16] shows how max flow—min cut duality leads to an elegant proof of Cheeger’s inequality, giving the lower bound in (1.18) on the first eigenvalue of the Laplacian on Ω. The eigenfunction has u = 0 on ∂Ω, so ∂ΩN is empty: Cheeger: λ1 ≥

1 2 h 4

where h(Ω) = tmax with F ≡ 1.

(1.7)

The Cheeger constant h is found from the constrained isoperimetric problem that arises for the minimal cut ∂S: Definition: h(Ω) = inf

S⊂Ω

perimeter of S . area of S

(1.8)

As in the particular case of our challenge problem, h(Ω) is often computable. For the unit square we note in (1.24) that the inequality (1.7) is far from tight.

2. Appleton and Talbot [2] have proposed an algorithm for computing the maximum flow vector v from a sequence of discrete problems. Their motivation is to study image segmentation with medical applications (see especially [3, 4]). The same techniques are successful in stereo matching [26]. Their paper is rich in ideas for efficient computations and an excellent guide to the literature.

4

Gilbert Strang

The algorithm approaches the maximum flow field as T → ∞, by introducing a “Maxwell wave equation” with capacity c and internal source F = 0: Appleton—Talbot:

∂E = − div v, ∂T

∂v = − grad E, ∂T

|v| ≤ c.

(1.9)

The potential is E, and the first equation ∂E/∂T = − div v is a relaxation of the conservation constraint v = 0 (Kirchhoff’s law). Appleton and Talbot RR div prove that the energy (|E|2 + |v|2 ) is decreasing in every subset S of Ω. At convergence, the optimal cut is the boundary of a level set of E. The equations (1.9) are discretized on a staggered grid. This corresponds to Yee’s method (also called the FTDT method) in electromagnetics. The algorithm has a weighting function to model the effect of source terms, and the experiments with image segmentation are very promising. Because primal—dual interior point algorithms have become dominant in optimization, we conjecture that those methods can be effective also here in the approximation of continuous by discrete maximal flows.

3. Nozawa [23] took a major step in extending the max flow—min cut theorem from the simple isotropic condition |v| ≤ 1 in (1.4) toward the much more general capacity condition (1.5). This step can be illustrated already in our challenge problem, by changing from the 2 norm of v(x, y) to the 1 or ∞ norm: 1

challenge:

∞

Maximize t so that div v = t with |v1 | + |v2 | ≤ 1 in Ω. (1.10)

challenge: Maximize t so that div v = t with |v1 | ≤ 1, |v2 | ≤ 1 in Ω. (1.11)

In the isoperimetric problem (1.8), this changes the definition of perimeter. The dual norm (in this case ∞ or 1 ) becomes the measure of arclength (dx, dy). Then this dual norm enters the computation of |∂S|: Z 2 Perimeter (in R ): |∂S| = |(dx, dy)|. (1.12) ∂S

The coarea formula from geometric measure theory [12], on which the proof of duality rests, continues to apply with the new definition. As in the 2 case, the maximal t can be computed! So we have new flow fields to find, reaching bounds that duality says are achievable. It is intriguing to connect maximal flow with the other central problem for networks and continua: the transportation problem. This asks for shortest


5

paths. The original work of Monge and Kantorovich on continuous flows has been enormously extended by Evans [11], Gangbo and McCann [15], Rachev and R¨ uschendorf [25], and Villani [33]. Our challenge problem requires the movement of material F (x, y) from Ω to ∂Ω. The bottleneck is in moving from the interior of S to the minimal cut ∂S. The distribution of material is uniform in S, and its destination is uniform along ∂S, to use all the capacity allowed by |v| ≤ 1. How is the shortest path (Monge) flow from S to ∂S related to the maximum flow?

4. Directed Graphs and Flows. Chung [8, 9] has emphasized that Cheeger’s theory (and the Laplacian itself) is not yet fully developed for directed graphs. For maximal flow on networks, Ford and Fulkerson [13] had no special difficulty when the edge capacities depend on the direction of flow. The problem is still a linear program and duality still holds. For directed continuous flows we lack a correctly formulated duality theorem. The capacity would be a constraint v(x, y) ∈ K(x, y) as in (1.5). Nozawa’s duality theorem in [23] quite reasonably assumed that zero is an interior point of K. Then a flow field exists for sufficiently small t (the feasible set is not empty). The continuous analogue of direction-dependent capacities seems to require analysis of more general convex sets K(x, y), when zero is a boundary point. In [22], Nozawa illustrated duality gaps when his hypotheses were violated. Finally we mention that all these questions extend to domains Ω in Rn . The constrained isoperimetric problems generalize to higher dimensions as well as different norms. The one simplification in the plane is the introduction of a stream function s(x, y), with (v1 , v2 ) = (∂s/∂y, −∂s/∂x) as the general solution to div v = 0. Our survey [30] formulated the corresponding primal and dual problems for s(x, y) as L1 and L∞ approximations of planar vector fields, where Laplace’s equation corresponds to L2 . The remaining sections of this chapter discuss the topics outlined above. We compute the minimum cuts in the three versions of the challenge problem on the unit square. We also mention an isoperimetric problem (with a different definition of perimeter) to which we return in a later paper [32].

1.3 Duality, Coarea, and Cheeger Constants The maximum flow is limited by the capacity c(x, y):

6

Gilbert Strang

Primal problem: Maximize t subject to div v = tF (x, y) in Ω, v · n = tf (x, y) on ∂ΩN , |v(x, y)|2 ≤ c(x, y) in Ω. Nozawa’s duality theorem requires a proper choice of function spaces and boundary conditions, in this problem and in its dual for u(x, y) in BV(Ω). Where the primal involves the divergence, the dual RR involves the RR gradient. Kohn and Temam [19] extended Green’s formula u div v = − v · grad u to allow functions u(x, y) of bounded variation. We show that the optimal u(x, y) in the dual problem is the characteristic function of a set S with finite perimeter. This u(x, y) is not smooth, but it lies in BV. The dual problem does not initially ask for a minimum cut. kukBV,c Dual problem: Minimize kukBV,c with (u) = 1 or Minimize | (u)| ZZ Z ZZ kukBV,c = c(x, y)| grad u|2 dx dy (u) = uf ds − uF dx dy. Ω

Ω

∂ΩN

(1.13) The key step toward the solution is to recognize RR the extreme points of the unit ball in this weighted BV norm kukBV,c = c | grad u|2 dx dy. Those extreme points are characteristic functions u = χS of open subsets S of Ω: χS (x, y) = {1 for x, y in S, 0 otherwise}. R The BV norm of χS is the weighted perimeter c ds of S, because the gradient is a measure (a line of delta functions) supported only on that boundary ∂S. The coarea formula gives the BV norm of u (weighted by the capacity c) as an integral over the norms of characteristic functions of level sets S(t) of u: Z Coarea: kukBV,c = kχS(t) kBV,c dt with S(t) = {x, y | u(x, y) < t}.

(1.14) Consider the case with F ≥ 0 and no boundary sources f . Specializing in (1.13) to the characteristic functions u = χS , our dual problem reduces to an isoperimetric problem for S and the minimum cut ∂S: R RR weighted perimeter ∂S c ds c | grad u|2 dx dy RR RR min . (1.15) = min u ∈ BV S ⊂ Ω weighted area F dx dy F u dx dy S Choosing c(x, y) = 1 and F (x, y) = 1, this computes the Cheeger constant. Cheeger constant: h(Ω) = inf

S ⊂Ω

|∂S| . |S|

(1.16)

R RR Weak duality h ≥ t is the inequality c ds ≥ t F dx dy for every feasible t in the primal problem. This is just the divergence theorem when div v =


7

tF (x, y) and |v| ≤ 1: Z

Weak duality h ≥ t:

c ds ≥

∂S

Z

∂S

v·n ds =

ZZ S

div v dx dy = t

ZZ

F dx dy. (1.17)

S

Duality says that equality holds for the maximal flow v and the minimal cut ∂S. Historically, the key inequality given by Cheeger [7] was a lower bound on the first eigenvalue λ1 of the Laplacian on the domain Ω. Grieser [16] observed how neatly and directly this bound follows from Green’s formula, when F = 1 and |v| ≤ 1. We expect to see the Schwarz inequality in the step from problems in L1 and L∞ to the eigenvalue problem in L2 : ZZ Z ZZ t u2 = (div v)u2 = − v(grad u2 ) ≤2

ZZ

∙Z Z ¸1/2 ZZ |u| | grad u| ≤ 2 u2 . | grad u|2

Thus any feasible t gives a lower bound t2 /4 to the Rayleigh quotient for any u(x, y) with u = 0 on ∂Ω: RR | grad u|2 dx dy t2 RR . (1.18) ≤ 4 u2 dx dy

The minimum of the right side is λ1 (Ω), and the maximum of the left side is h2 /4. Cheeger’s inequality becomes h2 /4 ≤ λ1 (Ω). A widely studied paper [10] of Diaconis and Stroock introduces another very useful measure of the “bottleneck” that limits flow on a graph.

1.4 The Challenge Problems When we use the 2 norm of the flow vector v = (v1 , v2 ) at each point, the constraint |v(x, y)| ≤ c(x, y) is isotropic. Other norms of v give constraints that come closer to those on a discrete graph. The edges of the graph might be horizontal and vertical (from a square grid) or at 45◦ and −45◦ (from a staggered grid). We use the challenge problem with F = c = 1 on a unit square as an example that allows computation of the minimal cut in all three cases. 2

constraint:

1

constraint: constraint:

∞

v12 + v22 ≤ 1.

|v1 | + |v2 | ≤ 1. max(|v1 |, |v2 |) ≤ 1.

(1.19) (1.20) (1.21)

8

Gilbert Strang

The 1 and ∞ norms give problems in linear programming. 2 The dual (minimum we RR cut) problems use the dual norms. For R had the usual BV norm | grad u| dx dy and the usual measure |∂S| = ds of the perimeter. For the 1 and ∞ constraints, the BV norms change and the perimeters reflect those changes (coming from the coarea formula in the new norms): |v|1 ≤ 1

|v|∞ ≤ 1

leads to ZZ | grad u|∞ dx dy kukBV = leads to ZZ | grad u|1 dx dy kukBV =

and |∂S|∞ =

Z

max(|dx|, |dy|),

∂S

and |∂S|1 =

Z

|dx| + |dy|.

∂S

The perimeter of a square changes as the square is rotated, because the norm of (dx, dy) is changing. In each case the dual problem looks for the minimum cut as the solution to a constrained isoperimetric problem. Duals of 2 , 1, ∞ :

Minimize 2 S ⊂ [0,1]

|∂S| |S|

and

|∂S|∞ |S|

and

|∂S|1 . |S|

(1.22)

In all cases the optimal S will reach the boundary ∂Ω of the square. (If S is stretched by a factor c, the areas in the denominators of (1.22) are multiplied by c2 and the numerators by c.) The symmetry of the problem ensures that the optimal ∂S contains four flat pieces of ∂Ω, centered on the sides of the square (Figure 1.1). The only parameter in the three optimization problems is the length L of those boundary pieces. Figure 1.1a shows the solution for the 2 problem, where the “unconstrained” parts of the cut ∂S are circular arcs. This follows from the classical isoperimetric problem, and it is easy to show that the arcs must be tangent to the square. The four arcs would fit together in a circle of radius r. With L = 1 − 2r, the optimal cut solves the Cheeger problem: tmax = h(Ω) = min

perimeter of S 4(1−2r) + 2πr . = min area of S 1 − 4r2 + πr2

(1.23)

The derivative of that ratio is zero when (1 − 4r2 + πr2 )(8 − 2π) = (4 − 8r + 2πr)(8r − 2πr).

√ Cancel 8 − 2π to reach 1 − 4r + (4 − π)r2 = 0. Then r = 1/(2 √ + π) ≈ .265. The Cheeger constant h(Ω) is the ratio |∂S|/|S| = 1/r = 2 + π. A prize of √ 10,000 yen was offered in [30] for the flow field that achieves div v = 2 + π with |v| ≤ 1. Lippert [20] and Overton [24] have the strongest

1 Maximum Flows and Minimum Cuts in the Plane r

S

L = 1−2r

R

S

L = 1−2R

r |∂S| = 4L + 2πr

9

S

L=1

R |∂S|∞ = 4L + 4R

Fig. 1.1 The minimum cuts ∂S for

2, 1,

and

∞

|∂S|1 = 4 constraints on v(x, y).

claim on the prize, by computing a close approximation to v. The discrete velocity clearly confirms the cut in Figure 1.1a as the set where |v| = 1. The eigenfunctions of the Laplacian on the unit square are (sin πx)(sin πy) and the lowest eigenvalue is λ1 = 2π2 . Cheeger’s inequality λ1 ≥ h2 /4, which other authors have tested earlier, is far from tight. √ Unit square: 2π 2 > (2 + π)2 /4 or 19.74 > 3.56 . (1.24) The second challenge problem has |v1 | + |v2 | ≤ 1 leading to the measure |∂S|∞ of the perimeter in the dual. Now the unconstrained isoperimetric √ problem is solved by a diamond with |n1 | = |n2 | = 1/ 2 on all edges. The optimal cut ∂S in Figure pieces and diamond √ 1.1b is a union of boundary √ edges. The edge length 2R is multiplied by 1/ 2 from |n1 | = √ |n2 | to give |∂S|∞ = 4L + 4R = 4 − 4R. Then the minimum cut has R = (2 − 2)/2 ≈ .3: √ |∂S|∞ 4 − 4R 2 =√ ≈ 3.5 . min = min R 1 − 2R2 |S| 2−1 For the flow field v in this 1 problem, the prize is reduced to 5000 yen. Lippert has reached the computational prize also in 1 . This is linear programming and interior point methods soundly defeated the simplex method. We cannot afford the prize in the ∞ problem, whose solution is simply v = (2x − 1, 2y − 1) on the square 0 ≤ x, y ≤ 1 with div v = 4 = tmax . The minimum cut for the ∞ problem is the whole boundary of the square. This coincides with the R unconstrained isoperimetric solution when the perimeter is measured by |dx| + |dy|. The minimizing set S would have horizontal and vertical sides wherever the constraint S ⊂ Ω is inactive, and here it is active everywhere on ∂S = ∂Ω. The Cheeger constant in this norm is h = 4/1. In [32] we prove that the unit ball in the dual norm (rotated by π/2) is isoperimetrically optimal. Here that ball is a circle or a diamond or a square. This isoperimetrix was discovered by Busemann [5] using the Brunn— Minkowski theorem in convex geometry (the Greeks knew much earlier about the circle). Our proof finds a simple linear equation for the support function of the optimal convex set S.

10

Gilbert Strang

Acknowledgment This research was supported by the Singapore—MIT Alliance. Added in proof Z. Milbers (unpublished thesis, K¨ oln Universit¨ at, 2006) has found the flow field in our challenge example (described at the end of Section 1.1)! New applications of minimum cuts and maximum flow have also appeared in landslide modeling, the L1 Laplacian, and especially image segmentation.

References 1. N. Alon, Eigenvalues and expanders, Combinatorica 6 (1986) 86—96. 2. B. Appleton and H. Talbot, Globally minimal surfaces by continuous maximal flows, IEEE Trans. Pattern Anal. Mach. Intell. 28 (2006) 106—118. 3. Y. Boykov and V. Kolmogorov, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision, IEEE Trans. Pattern Anal. Mach. Intell. 26 (2004) 1124—1137. 4. Y. Boykov, O. Veksler, and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Trans. Pattern Anal. Mach. Intell. 23 (2001) 1222—1239. 5. H. Busemann, The isoperimetric problem in the Minkowski plane, Amer. J. Math. 69 (1947) 863—871. 6. J. D. Chavez and L. H. Harper, Duality theorems for a continuous analog of FordFulkerson flows in networks, Adv. Appl. Math. 14 (1993) 369—388. 7. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, 1970, 195—199. 8. F. R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, 1997. 9. F. R. K. Chung, Laplacians and the Cheeger inequality for directed graphs, Ann. Combinatorics 9 (2005) 1—19. 10. P. Diaconis and D. W. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab. 1 (1991) 36—61. 11. L. C. Evans, Survey of applications of PDE methods to Monge-Kantorovich mass transfer problems, www.math.berkeley.edu/∼evans (earlier version: Current Developments in Mathematics, 1997). 12. W. Fleming and R. Rishel, An integral formula for total gradient variation, Archiv der Mathematik 11 (1960) 218—222. 13. L. R. Ford Jr. and D. R. Fulkerson, Flows in Networks, Princeton University Press, 1962. 14. L. R. Ford Jr. and D. R. Fulkerson, Maximal flow through a network, Canad. J. Math. 8 (1956) 399—404. 15. W. Gangbo and R. McCann, Optimal maps in Monge’s mass transport problem, C.R. Acad. Sci. Paris. Ser. I. Math. 325 (1995) 1653—1658. 16. D. Grieser, The first eigenvalue of the Laplacian, isoperimetric constants, and the max flow min cut theorem, Archiv der Mathematik 87 (2006) 75—85. 17. T. C. Hu, Integer Programming and Network Flows, Addison-Wesley, 1969. 18. M. Iri, Theory of flows in continua as approximation to flows in networks, Survey of Mathematical Programming 2 (1979) 263—278. 19. R. Kohn and R. Temam, Dual spaces of stresses and strains, Appl. Math. and Opt. 10 (1983) 1—35. 20. R. Lippert, Discrete approximations to continuum optimal flow problems, Stud. Appl. Math. 117 (2006) 321—333. 21. J. S. Mitchell, On maximum flows in polyhedral domains, Proc. Fourth Ann. Symp. Computational Geometry, 341—351, 1988.


11

22. R. Nozawa, Examples of max-flow and min-cut problems with duality gaps in continuous networks, Math. Program. 63 (1994) 213—234. 23. R. Nozawa, Max-flow min-cut theorem in an anisotropic network, Osaka J. Math. 27 (1990) 805—842. 24. M. L. Overton, Numerical solution of a model problem from collapse load analysis, Computing Methods in Applied Science and Engineering VI, R. Glowinski and J. L. Lions, eds., Elsevier, 1984. 25. S. T. Rachev and L. R¨ uschendorf, Mass Transportation Problems I, II, Springer (1998). 26. S. Roy and I. J. Cox, A maximum-flow formulation of the n-camera stereo correspondence problem, Proc. Int. Conf. Computer Vision (1988) 492—499. 27. F. Santosa, An inverse problem in photolithography, in preparation. 28. R. Sedgewick, Algorithms in C, Addison-Wesley, 2002. 29. G. Strang, A minimax problem in plasticity theory, Functional Analysis Methods in Numerical Analysis, Z. Nashed, ed., Lecture Notes in Mathematics 701, Springer, 1979. 30. G. Strang, L1 and L∞ approximation of vector fields in the plane, Lecture Notes in Num. Appl. Anal. 5 (1982) 273—288. 31. G. Strang, Maximal flow through a domain, Math. Program. 26 (1983) 123—143. 32. G. Strang, Maximum area with Minkowski measures of perimeter, Proc. Royal Society of Edinburgh 138 (2008) 189—199. 33. C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Mathematical Society, 2003.

Chapter 2

Variational Principles and Residual Bounds for Nonpotential Equations Giles Auchmuty Dedicated to Gil Strang for his 70th birthday

Summary. Solutions of nonsymmetric linear equations whose symmetric part is positive definite are first characterized as saddle points of a strictly convex—concave quadratic function. The associated primal problem is shown to be equivalent to a weighted quadratic minimum residual optimization problem. An a posteriori error estimate for approximate solutions is derived. Similar results are then obtained for semilinear finite-dimensional systems of equations. These include global optimization problems for the solutions and existence results based on min-max theorems. Under further assumptions, uniqueness theorems are proven using saddle point theorems.

2.1 Introduction In this chapter, some unconstrained global optimization problems for the solutions of finite-dimensional systems of equations that are not of potential type are described and analyzed. That is, the equations need not be obtained directly as the critical points of a differentiable function. This approach is first illustrated by considering the case of a linear equation involving a matrix that is nonsymmetric but has positive definite symmetric part. In Section 2.2 the solutions of such equations are shown to be the saddle point of certain functions, and the existence and uniqueness of these saddle points are proven directly. The dual variational principles associated with this saddle point problem are studied. In Section 2.3, the primal problem associated with this saddle problem is shown to be a weighted quadratic Giles Auchmuty Department of Mathematics, University of Houston, Houston, TX 77204-3008 U.S.A. e-mail: [email protected] This work was partially performed while the author was working for the Division of Mathematical Sciences, National Science Foundation, Arlington, VA. D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_2, © Springer Science+Business Media, LLC 2009

13

14

Giles Auchmuty

residual minimization problem. Thus the variational principle provides elementary error estimates for the solution in terms of the value of the function being minimized. These results extend to nonsymmetric problems some of the results that are well known for the solutions of symmetric positive definite problems. See the recent texts of Ainsworth and Oden [1] and Han [7] for applications of such results to the analysis of finite element simulations. Sections 2.4 and 2.5 describe extensions of these results to two very different classes of semilinear finite-dimensional equations. In each case, a min-max characterization of the solutions is described. These need not be saddle point problems but there is a primal functional whose global minima provide solutions of the original problem. A generalized Young’s inequality and results from minimax theory are used to prove existence results and obtain characterizations of the solutions. The problem in Section 2.4 may again be interpreted as a minimum residual problem and residual bounds for approximate solutions are described. The problem in Section 2.5 uses similar methods but makes quite different assumptions on the nonlinear term to obtain existence and, under further assumptions, uniqueness results. It is a particular pleasure to contribute this chapter to these conference proceedings in honor of Gilbert Strang as essentially all of the background material for these results has been beautifully treated in his texts on linear algebra and applied mathematics.

2.2 Saddle Point Characterizations of Nonsymmetric Linear Equations Let A be an n × n real matrix that satisfies the coercivity (or ellipticity) condition that there is an a0 > 0 such that hAx, xi ≥ a0 kxk2

for all x ∈ Rn .

(2.1)

Here the brackets indicate the usual Euclidean inner product and the norm is the usual 2-norm. In general a function V : Rn → R is said to be coercive provided kxk−1 V (x) → ∞ as kxk → ∞. In this section, it is shown that, when A is coercive, the solutions of the linear equation Ax = f (2.2) for given f ∈ Rn , may be characterized as saddle points of a quadratic convex—concave function. When A is real symmetric and coercive, it is well known that the solution of (2.2) is the unique minimizer of the energy functional E(x) on Rn defined by

2 Variational Principles and Residual Bounds for Nonpotential Equations

E(x) := hAx, xi − 2hf, xi.

15

(2.3)

A variety of different aspects of this problem is treated in Strang [8]. When A is coercive, but not real symmetric, then Auchmuty [3] showed that the solution of (2.2) may be characterized as the saddle point of a convex—concave function. Here a different saddle function is used that provides minimum residual variational principles and computable error bounds for approximate solutions of (2.2). Let AS := (A + AT )/2 and B := (A − AT )/2 be the symmetric and skew symmetric parts of A, D be a diagonal positive definite matrix, and C := AS − D. Then C is a real symmetric matrix and equation (2.2) may be written (B + C + D) x = f. (2.4) When C1 , C2 are real symmetric matrices, we say that C2 ≤ C1 (respectively, C2 < C1 ) provided h(C1 − C2 )x, xi ≥ 0 for all x ∈ Rn , (or h(C1 − C2 )x, xi > 0 for all nonzero x ∈ Rn ). Consider the function L : Rn × Rn → R defined by L(x, y) := h(AS −D/2)x, xi + hf, y−xi − h(B+C)x, yi −

1 hDy, yi. (2.5) 2

A point (ˆ x, yˆ) ∈ Rn × Rn is said to be a saddle point of L provided L(ˆ x, y) ≤ L(ˆ x, yˆ) ≤ L(x, yˆ)

for all (x, y) ∈ Rn × Rn .

The following result shows that, provided D is small enough, the function L defined above has a unique saddle point, and that this saddle point characterizes the solutions of the linear equation (2.2). Theorem 2.1. Assume A, B, C, D as above, (2.1) holds, and D < 2AS . Then the function L defined by (2.5) has a unique saddle point (ˆ x, x ˆ) ∈ Rn × Rn with x ˆ being the unique solution of (2.2). Proof. Given y in Rn , L(., y) is continuous. When D < 2AS , this function is strictly convex and there is a δ > 0 and a function c such that L(x, y) ≥ δkxk2 − [kf k + k(C − B)yk] kxk + c(y). Hence L(., y) is coercive for each y ∈ Rn . Similarly −L(x, .) is continuous, strictly convex, and coercive for each x ∈ Rn . The usual minimax theorem (see Theorem 49A in Zeidler [9] for example) implies that L has a saddle point in Rn × Rn . This function L is continuously differentiable, so the saddle point is a solution of the system ∇x L(x, y) = (2AS − D)x + (B − C)y − f = 0 ∇y L(x, y) = f − (B + C)x − Dy = 0.

(2.6) (2.7)

16

Giles Auchmuty

Here ∇ is the usual gradient operator. Add these equations to obtain (C + D − B)(x − y) = 0. Take inner products with (x − y); then hAS (x − y), x − yi ≥ a0 kx − yk2 = 0 as AS = C + D, (2.1) holds, and hBz, zi = 0 for all z. Thus the saddle point must have the form (ˆ x, x ˆ) and (2.7) shows that the equation satisfied by x ˆ is (2.2). The uniqueness of solutions follows from (2.1). t u Note that this is a (quite different) existence-uniqueness proof for (2.2) when (2.1) holds. It is based on elementary calculus and the minimax theorem. The analysis in Auchmuty [3] used a different saddle function that did not involve a matrix of the form D. The introduction of this matrix leads to expressions that are more practical for numerical computation than those described in [3]. Strictly speaking, it is not necessary that D be diagonal; it suffices that D be symmetric and positive definite and that the inverse D−1 be known explicitly for use in formulae to be described in the next section.

2.3 Variational Principles for Nonsymmetric Linear Equations A convex—concave saddle problem defines a pair of associated dual variational principles. See Auchmuty [2], Section 3 or Ekeland and Temam [5], Chapter III for general descriptions of such constructions and their properties. Define the function G : Rn → R by G(x) := sup L(x, y). n y∈R

(2.8)

The primal problem associated with L is to minimize G on Rn . The maximization of L with respect to y is straightforward and yields 1 1 h(AS −D/2)x, xi − hf, xi + hD−1 (f −(B +C)x), f −(B +C)xi. 2 2 (2.9) The essential results about this minimization problem may be summarized as follows. G(x) =

Theorem 2.2. Assume A, B, C, D as above and (2.1) holds. Then the function G defined by (2.9) is strictly convex and coercive on Rn . It has a unique minimizer x ˆ on Rn with x ˆ being the unique solution of (2.2). Proof. First note that D−1 is diagonal with positive entries on the diagonal, as this holds for D. Straightforward algebra leads to the formula


1 hD−1 (f − Ax), f − Axi 2 1 1 = hD−1 Ax, Axi − hAT D−1 f, xi + hD−1 f, f i. 2 2

G(x) =

17

(2.10) (2.11)

From (2.1) and Cauchy’s inequality one has kAxk ≥ a0 kxk. Let dM be the largest entry in the diagonal matrix D; then G(x) ≥

a0 kxk2 − c1 kf k kxk + c2 2dM

for all x ∈ Rn

(2.12)

where c1 , c2 are constants. These formulae imply that G is strictly convex and coercive on Rn as it is quadratic. It is continuous so G attains its infimum and the minimizer is unique. The expression (2.11) is G-differentiable with ∇G(x) = AT D−1 (Ax − f ). This must be zero at a minimizer. Because A and D are nonsingular, this implies that the minimizers satisfy (2.2) as claimed. t u This function G was defined from general considerations of duality associated with the function L. Let r(x) := f − Ax be the residual of x with respect to the equation (2.2); then (2.11) implies that G(x) =

1 1 kr(x)k2 hD−1 r, ri ≥ 2 2dM

for all x ∈ Rn .

(2.13)

This shows that the variational principle of minimizing G on Rn is a weighted (or preconditioned) minimum residual principle for the problem of solving (2.2). This enables us to obtain an a posteriori error estimate for solutions of (2.2) in terms of the values of G(x). When x ˆ is the solution of (2.2) and x ∈ Rn , then A(x − x ˆ) = r, so upon taking inner products with x−x ˆ and using (2.1) one sees that a0 kx − x ˆk2 ≤ |hr, x − x î| ≤ krk kx − x ˆk Rearrange this; then (2.13) implies kx − x ˆk ≤ a0 −1 krk ≤ a0 −1

p 2dM G(x)

for all x ∈ Rn .

(2.14)

This estimate does not require knowledge of a condition number for A; just a bound on a0 . Note that D here is any positive definite diagonal matrix, but the values of G(x) depend on both D, D−1 . This suggests that in practice one may wish to investigate the dependence of these bounds on the choice of D for a particular matrix A in (2.2). The dual problem associated with the saddle point problem for L is to maximize H : Rn → R defined by H(y) := inf x∈Rn L(x, y).

(2.15)

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Giles Auchmuty

The explicit formula for H analogous to (2.9) is 1 1 H(y) = − h(2C + D)−1 (f + (C − B)y), f + (C − B)yi + hf, yi − hDy, yi. 2 2 (2.16) The use of this function requires the evaluation of (2C + D)−1 which usually is more effort than the determination of D−1 , so we just concentrate on use of the primal problem. This dual functional has very similar properties to −G.

2.4 Variational Principles for Semilinear Equations I The preceding analysis generalizes to large classes of semilinear finite-dimensional systems of equations. First consider an equation of the form Ax = F (x),

(2.17)

where A is a real n × n matrix which is coercive but need not be symmetric, and F : Rn → Rn is a continuous function. Let B, C, and D be matrices defined as before, so that this equation may be written as F (x) − (B + C)x = Dx = ∇q(x)

with q(x) :=

1 hDx, xi. (2.18) 2

Consider the function G : Rn → R defined by G(x) :=

1 1 hDx, xi + hD−1 (F (x) − (B + C)x), F (x) − (B + C)xi 2 2 + hCx − F (x), xi (2.19)

and the variational problem of minimizing G on Rn . Let the value of this problem be α(G) := inf n G(x). x∈R This is a generalization of the variational principle described in Section 2.3 with F (x) replacing f . The essential properties of this optimization problem may be summarized as follows. Theorem 2.3. Assume A, B, C, D, and F as above. Then the function G defined by (2.19) is continuous and has value α(G) ≥ 0. A point x ˆ ∈ Rn is n a solution of (2.17) if and only if x ˆ minimizes G on R and G(ˆ x) = 0. Proof. Let q(x) be the quadratic form defined in (2.18); then q is strictly convex on Rn and its conjugate function is q ∗ (z) := 12 hD−1 z, zi. Then from the generalized Young inequality, (see Proposition 51.2 in Zeidler [9] or Section 2.5 in Han [7]), one sees that


19

for all x ∈ Rn . (2.20) This and the fact that B is skew-symmetric implies that G(x) ≥ 0 for all x ∈ Rn . Hence α(G) ≥ 0. Equality holds in (2.20) if and only if q(x) + q ∗ (F (x) − (B + C)x) − hF (x) − (B + C)x, xi ≥ 0

F (x) − (B + C)x = Dx. This is equation (2.17) and at such a point G(x) = 0.

t u

This result provides a variational principle for the possible solutions of our equation that requires not only that the points be global minimizers but also that the value of the problem be zero. It is a straightforward computation to verify that the analogue of equation (2.13) remains valid for this nonlinear problem. Specifically let r(x) := Ax − F (x) be the residual of this equation at a point x ∈ Rn , then G(x) =

1 1 kr(x)k2 hD−1 r(x), r(x)i ≥ 2 2dM

for all x ∈ Rn .

(2.21)

Thus the function G is again a weighted residual function for this problem and G(x) small implies that the residual is small. Without further conditions on the nonlinearity F there is no guarantee that the residual being small implies that a point x is close to a solution of the orginal equation. To obtain such a condition on the function F we use an associated min-max problem. Consider the function M : Rn × Rn → R defined by 1 hDy, yi. 2 (2.22) A point (ˆ x, yˆ) ∈ Rn × Rn is said to be a min-max point of M provided

M(x, y) := h(AS − D/2)x, xi + hF (x), y − xi − h(B + C)x, yi −

M(ˆ x, yˆ) =

inf n sup M(x, y). n y∈R

x∈R

(2.23)

Obviously a saddle point of M is a min-max point but the converse need not hold; see Auchmuty [4], Section 2 for descriptions of this. First note that this expression for M implies that G(x) = sup M(x, y), n y∈R

(2.24)

so (ˆ x, yˆ) is a min-max point of M implies that x ˆ is a minimizer of G. Here, and in the next section, we need a general result about the existence of such min-max points. The theorem that is used may be stated as follows. Theorem 2.4. Let K be a nonempty closed convex set in Rn and assume that M : K × K → R satisfies (i) M(x, x) ≤ 0 for all x in K.

20

Giles Auchmuty

(ii) For each x in K, M(x, .) is concave on K. (iii) For each y in K, M(., y) is l.s.c. on K. (iv) For some y0 in K, the set E0 := {x ∈ K : M(x, y0 ) ≤ 0} is bounded in K. Then there is an x0 ∈ K satisfying supy∈K M(x0 , y) ≤ 0. This result is a specialization of Theorem 3 in Auchmuty [4] and is esssentially due to Ky Fan [6]. Theorem 2.5. Suppose A, B, D, and F as above and the set of points that satisfy h(AS − D/2)x, xi − hF (x), xi ≤ 0 (2.25)

is bounded in Rn . Then M has a min-max point (ˆ x, yˆ) with G(ˆ x) = 0 and x ˆ is a solution of (2.17). Proof. This is proved using Fan’s theorem with K = Rn and M defined by (2.22). Condition (i) holds with M(x, x) = 0 as AS − D/2 = C + D/2 and B is skew-symmetric; (ii) holds as D is positive definite; and (iii) holds as M is continuous. Take y0 = 0 in condition (iv); then the criterion for E0 to be bounded is that the set of points for which (2.25) holds is bounded. Thus Theorem 2.4 yields that there is an x ˆ ∈ Rn for which G(ˆ x) ≤ 0. From Theorem 2.3, G(x) ≥ 0 for all x, so there is a minimizer of G with G(ˆ x) = 0 and it is a solution of (2.17). t u Suppose the nonlinear mapping F satisfies the following condition. Condition C1: There is an R > 0 and an a1 such that a1 < a0 and hF (x), xi ≤ a1 kxk2

for all kxk ≥ R.

(2.26)

This condition holds both in the linear case where F (x) = f is constant or when F is nonlinear and bounded on Rn . Corollary 2.6. Assume A satisfies (2.1), F is continuous and C1 holds. Then there is at least one solution x ˆ of (2.17) and x ˆ minimizes G on Rn . Proof. When (2.26) holds then the set of points for which (2.25) holds is bounded provided D is chosen to be sufficiently small. Thus Theorem 2.5 yields this result. t u

2.5 Variational Principles for Semilinear Equations II There is another class of semilinear, but not necessarily potential, equations for which there are useful variational principles whose minima provide solutions of the equations.


21

When a function V is continuous and coercive on Rn , then it is bounded below on Rn and its conjugate function V ∗ : Rn → R defined by V ∗ (z) := sup [hz, xi − V (x)] n x∈R is convex, lower semicontinuous, and finite for all z ∈ Rn . Consider the problem of solving equations of the form Ax + ∇V (x) = 0,

(2.27)

where A is a real n×n matrix and V is assumed to be C 1 and coercive on Rn . This may be regarded as a special case of the system treated in the previous section with F (x) = −∇V (x) but now A need not be coercive. Define J : Rn → R by J (x) := V (x) + V ∗ (−Ax) + hAS x, xi.

(2.28)

When V , A are as above then this function is finite for each x ∈ Rn and lower semicontinuous (l.s.c.). Consider the problem of minimizing J on Rn and finding α(J ) := inf n J (x). x∈R This problem has similar properties to that of the problem described in Section 2.4. Theorem 2.7. Assume V matrix. Then the function finite at each x and α(J ) and only if it minimizes J

is C 1 and coercive on Rn and A is a real n × n J defined by (2.28) is lower semicontinuous and ≥ 0. A point x ˆ ∈ Rn is a solution of (2.27) if n on R and J (ˆ x) = 0.

Proof. The function V ∗ is finite-valued, convex, and lower semicontinuous from our assumptions on V and because it is the supremum of a family of such functions. Thus J is finite and l.s.c. on Rn . The generalized Young’s inequality implies that J (x) ≥ 0 for all x and that J (x) = 0 if and only if x satisfies −Ax ∈ ∂ V (x).

Here ∂ V (x) is the subdifferential of V at x. Because V is G-differentiable, ∂ V (x) = {∇V (x)}, so the result follows. t u Here again, J provides a variational principle for the solutions of (2.27) for which the minimizers of J must satisfy an extra condition to actually be a solution of the equation. This result did not require that V be convex. When V is also convex, then some existence results for this problem can be obtained using min-max methods. Consider the function W : Rn × Rn → R defined by W(x, y) := V (x) + hAS x, xi − hAx, yi − V (y).

(2.29)

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Giles Auchmuty

This function had the property that J (x) = sup W(x, y). n y∈R

(2.30)

The following theorem describes conditions on the function V that guarantee existence of a minimizer of J that obeys the criteria of Theorem 2.7.

Theorem 2.8. Suppose V is convex, C 1 , and coercive on Rn , and the set of points satisfying V (x) + hAS x, xi ≤ V (0) (2.31)

is bounded in Rn . Then W has a min-max point (ˆ x, yˆ) and J (ˆ x) = 0. If, in addition, A satisfies (2.1), then W is convex—concave, (ˆ x, yˆ) is a saddle point of W, x ˆ is the unique minimizer of J , and there is a unique solution of (2.27). Proof. To prove the first part of this, the conditions of Theorem 2.4 are verified for W. Our assumptions on V imply that (i), (ii), and (iii) all hold. Take y0 = 0 in (iv); then the criteria above guarantee that the set of points which satisfy (2.31) is bounded. Thus Theorem 2.4 says that there is an x ˆ such that J (ˆ x) ≤ 0. From Theorem 2.7, J (x) ≥ 0 for all x, so J (ˆ x) = 0. For a given x ˆ, the supremum of W(ˆ x, .) is attained as V is convex and coercive, hence there is a min-max point. When A satisfies (2.1), W(., y) is strictly convex on Rn for each y and the remaining results follow from the usual saddle point theorem. t u The condition (2.31) says that provided lim inf [ V (x) + hAS x, xi ] > V (0)

kxk→∞

(2.32)

then the equation (2.27) has at least one solution, without requiring coercivity of A. In this case, coercivity of A yields uniqueness of the solution and dual variational principles for the problem may be described. The results described here can be generalized to variational principles and solvability results for equations between a real reflexive Banach space X and its dual space X ∗ using similar constructions and proofs. The relevant saddle point theorems in [9], or the min-max theorems of [4] hold in this generality. When natural conditions are imposed on A, F so that these theorems hold, results analogous to those described here for the finite-dimensional case may be stated.

References [1] M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis, John Wiley & Sons, New York, 2000.


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[2] G. Auchmuty, Duality for Non-Convex Variational Principles, J. Diff. Eqns. 50 (1983), 80—145. [3] G. Auchmuty, Saddle Point Methods, and Algorithms, for Non-Symmetric Linear Equations, Numer. Funct. Anal. Optim. 16 (1995), 1127—1142. [4] G. Auchmuty, Min-Max Problems for Non-Potential Operator Equations, Optimization Methods in Partial Differential Equations (S. Cox and I. Lasiecka, eds.), Contemporary Mathematics, vol. 209, American Mathematical Society, Providence, 1997, pp. 19—28. [5] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976. [6] K. Fan, A Generalization of Tychonoff’s Fixed Point Theorem, Math. Ann. 142 (1961), 305—310. [7] W. Han, A Posteriori Error Analysis via Duality Theory, Springer Science, New York, 2005. [8] G. Strang, An Introduction to Applied Mathematics, Cambridge-Wellesley Press, 1986. [9] E. Zeidler, Nonlinear Functional Analysis and Its Applications III, Variational Methods and Optimization, Springer Verlag, New York, 1985.

Chapter 3

Adaptive Finite Element Solution of Variational Inequalities with Application in Contact Problems Viorel Bostan and Weimin Han

Summary. In this chapter, we perform a posteriori error analysis for the adaptive finite element solution of several variational inequalities, including elliptic variational inequalities of the second kind and corresponding quasistatic variational inequalities. A general framework for a posteriori error estimation is established by using duality theory in convex analysis. We then derive a posteriori error estimates of residual type and of recovery type, through particular choices of the dual variable present in the general framework. The error estimates are guaranteed to be reliable. Efficiency of the error estimators is theoretically investigated and numerically validated. Detailed derivation and analysis of the error estimates are given for a model elliptic variational inequality. Extensions of the results can be made straightforward in solving other elliptic variational inequalities of the second kind, and we present such an extension for a problem arising in frictional contact. Moreover, we use a quasistatic contact problem as an example to illustrate how to extend the a posteriori error analysis in solving time-dependent variational inequalities. Numerous numerical examples are included to illustrate the effectiveness of the a posteriori error estimates in adaptive solutions of the variational inequalities. Key words: A posteriori error estimation, adaptive finite element solution, elliptic variational inequality, quasistatic variational inequality, frictional contact, duality, reliability, efficiency

3.1 Introduction In this chapter, we present some theoretical and numerical results on a posteriori error estimation and adaptive finite element solution of elliptic Department of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A. e-mail: [email protected], [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization, Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_3, © Springer Science+Business Media, LLC 2009

25

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V. Bostan, W. Han

variational inequalities of the second kind, as well as corresponding quasistatic variational inequalities, especially those arising in frictional contact problems. The general framework for a posteriori error estimation of the chapter works for any Galerkin solutions of the variational inequalities. However, we specifically choose the finite element method for approximation, as the method today is the dominant numerical method for solving most problems in structural and fluid mechanics. It is widely applied to both linear and nonlinear problems. General mathematical theory of finite element methods can be found in [4, 26, 27, 58, 64], among others. The textbook [51] offers an easily accessible mathematical introduction of finite element methods, whereas the two recent textbooks [17, 18] provide deeper mathematical theory together with more recent and current research development such as the multigrid methods. Traditionally, convergence of finite element solutions is achieved through mesh refinement with the use of a piecewise low degree polynomial. Because h is usually used to denote the mesh size, the traditional finite element method is also termed the h-version finite element method. On the other hand, convergence of the method can also be achieved by using piecewise increasingly higher degree polynomials over relatively coarse finite element meshes, leading to the p-version finite element method. Detailed discussion of the p-version finite element method can be found in [66]. The p-version method is more efficient in areas where the solution is smooth, so it is natural to combine the ideas of the p-version and the h-version to make the finite element method very efficient on many problems. A wellknown result regarding the h-p-version finite element method is the exponential convergence rate for solving elliptic boundary value problems with corner singularities, under proper combinations of local polynomial degrees and element sizes. Comprehensive mathematical theory of the p-version and h-p-version finite element methods with applications in solid and fluid mechanics can be found in [63]. Mixed and hybrid finite element methods are often used in solving boundary value problems with constraints and higher-order differential equations. Mathematical theory of these methods can be found in [19, 62]. Several monographs are available on the numerical solution of Navier—Stokes equations by the finite element method (see, e.g., [33]). Theory of the finite element method for solving parabolic problems can be found in [67] and more recently in [68]. Finally, we list a few representative engineering books on the finite element method, [11, 50, 76, 77]. The reader is referred to two historical notes [57, 75] on the development of the finite element method. For practical use of a numerical method, one important issue is the assessment of the reliability and accuracy of the numerical solution. The reliability of the numerical solution hinges on our ability to estimate errors after the solution is computed; such an error analysis is called a posteriori error analysis. A posteriori error estimates provide quantitative information on the accuracy

3 Finite Element Solution of Variational Inequalities with Applications

27

of the solution and are the basis for the development of automatic, adaptive solution procedures. The research on a posteriori error estimation and adaptive mesh refinement for the finite element method began in the late 1970s. The pioneering work on the topic was done in [5, 6]. Since then, a posteriori error analysis and adaptive computation in the finite element method have attracted many researchers, and a variety of different a posteriori error estimates have been proposed and analyzed. In a typical a posteriori error analysis, after a finite element solution is computed, the solution is used to compute element error indicators and an error estimator. The element error indicator represents the contribution of the element to the error in the computation of some quantity by the finite element solution, and is used to indicate if the element needs to be refined in the next adaptive step. The error estimator provides an estimate of the error in the computation of the quantity of the finite element solution, and thus can be used as a stopping criterion for the adaptive procedure. Often, the error estimator is computed as an aggregation of the element error indicators, and one usually only speaks of error estimators. Most error estimators can be classified into residual type, where various residual quantities (residual of the equation, residual from derivative discontinuity, residual of material constitutive laws, etc.) are used, and recovery type, where a recovery operator is applied to the (discontinuous) gradient of the finite element solution and the difference of the two is used to assess the error. Error estimators have also been derived based on the use of hierarchic bases or equilibrated residual. Two desirable properties of an a posteriori error estimator are reliability and efficiency. Reliability requires the actual error to be bounded by a constant multiple of the error estimator, up to perhaps a higher-order term, so that the error estimator provides a reliable error bound. Efficiency requires the error estimator to be bounded by a constant multiple of the actual error, again perhaps up to a higher-order term, so that the actual error is not overestimated by the error estimator. The study and applications of a posteriori error analysis is a current active research area, and the related publications grow fast. Some comprehensive summary accounts can be found, in chronicle order, in [70, 1, 7]. Initially, a posteriori error estimates were mainly developed for estimating the finite element error in the energy norm. In the recent years, error estimators have also been developed for goal-oriented adaptivity. The goal-oriented error estimators are derived to specifically estimate errors in quantities of interest, other than the energy norm errors. Chapter 8 of [1] is devoted to such error estimators. The latest development in this direction is depicted in [10, 32]. Most of the work so far on a posteriori error analysis has been devoted to ordinary boundary value problems of partial differential equations. In applications, an important family of nonlinear boundary value and initial boundary value problems is that associated with variational inequalities, that is, problems involving either differential inequalities or inequality boundary con-

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ditions. Mechanics is a rich source of variational inequalities (see, e.g., [59]), and some examples of problems that give rise to variational inequalities are obstacle problems, contact problems, plasticity and viscoplasticity problems, Stefan problems, unilateral problems of plates and shells, and non-Newtonian flows involving Bingham fluids. An early comprehensive reference on the topic is [29], where many nonlinear boundary value problems in mechanics and physics are formulated and studied in the framework of variational inequalities. A concise introduction to the mathematical theory of some variational inequalities can be found in [54]. Numerical approximations of general variational inequalities are studied in detail in [34, 35]. Numerical methods for some variational inequalities arising in mechanics are the subject of [47, 48]. Mathematical analysis and numerical approximations of variational inequalities arising in contact mechanics are presented in [53] (for elastic materials) and [46] (for viscoelastic and viscoplastic materials). In [43, 44], elastoplasticity problems are formulated and analyzed in the form of variational inequalities. Although several standard techniques have been developed to derive and analyze a posteriori error estimates for finite element solutions to problems in the form of variational equations, they do not work directly for a posteriori error analysis of numerical solutions to variational inequalities due to the inequality feature of the problems. Nevertheless, numerous papers can be found on a posteriori error estimation of finite element solutions of obstacle problems, for example, [2, 25, 49, 55, 56, 69] (these papers consider numerical solutions on convex subsets of finite element spaces), as well as [31, 52] (these papers use a penalty approach for discrete solutions). Obstacle problems are so-called variational inequalities of the first kind; that is, they are inequalities involving smooth functionals and are posed over convex subsets. We also note that a posteriori error estimation is discussed in [12, 13, 65], although the derivations of the estimates in these papers are arguable. In the context of elastoplasticity with hardening, computable a posteriori error estimates are derived in [3, 20, 22] for the primal problem, which is a variational inequality of the second kind; that is, the inequality arises as a result of the presence of a nondifferentiable functional. These works deal extensively also with a priori estimates, and in the latter work a number of numerical examples are presented. Residual type error estimators were studied for an elliptic variational inequality of the second kind in [15, 16]. In this chapter, we derive and study some a posteriori error estimates for finite element solutions of elliptic variational inequalities of the second kind and corresponding quasistatic variational inequalities. The basic mathematical tool we use is the duality theory in convex analysis (cf. [30, 73]). The duality theory has been applied to derive efficient a posteriori error estimates for mathematical idealizations of physical and engineering problems (see, e.g., [37, 38]), as well as for some numerical procedures for solving nonlinear problems, such as the regularization techniques in [36, 42, 45], and the Kaˇcanov iteration method in [40, 41]. A summary account of these can be found in


29

[39]. In [61, 60], the technique of the duality theory was used to derive a posteriori error estimates of the finite element method in solving boundary value problems of some nonlinear equations. In these papers, the error bounds are shown to converge to zero in the limit; however, no efficiency analysis of the estimates is given. For convenience, we recall here a representative result on the duality theory (see [30]). Let V , Q be two normed spaces, and denote by V ∗ , Q∗ their dual spaces. Assume there exists a linear continuous operator Λ ∈ L(V, Q), with transpose Λ∗ ∈ L(Q∗ , V ∗ ). Let F be a functional mapping V × Q into the extended real line R ≡ R ∪ {±∞}. Consider the minimization problem: inf F (v, Λv).

(3.1)

sup [ −F ∗ (Λ∗ q ∗ , −q ∗ ) ] ,

(3.2)

v∈V

Define its dual problem by q ∗ ∈Q∗

where F ∗ is the conjugate functional of F : F ∗ (v∗ , q ∗ ) = sup [hv, v∗ i + hq, q ∗ i − F (v, q)] .

(3.3)

v∈V q∈Q

Then we have the following theorem. Theorem 3.1. Assume (1) (2) (3) (4) (5)

V is a reflexive Banach space and Q a normed space, F : V × Q → R is a proper, lower semicontinuous, convex function, Λ : V → Q is a linear bounded operator with its adjoint Λ∗ : Q∗ → V ∗ , ∃ u0 ∈ V with F (u0 , Λu0 ) < ∞ and q 7→ F (u0 , q) continuous at Λu0 , F (v, Λv) → +∞ as kvk → ∞ ∀ v ∈ V.

Then the problem (3.1) has a solution u ∈ V , its dual (3.2) has a solution p∗ ∈ Q∗ , and F (u, Λu) = −F ∗ (Λ∗ p∗ , −p∗ ). Furthermore, if F is strictly convex, then a solution u of problem (3.1) is unique. The rest of the chapter is organized as follows. In Section 3.2 we introduce a model elliptic variational inequality of the second kind and its finite element approximation. We provide detailed derivation and analysis of a posteriori error estimates of the finite element solutions for the model problem. In Section 3.3 we formulate a dual problem for the model, and use the dual problem to establish a general a posteriori error estimate for any approximation of the solution of the model elliptic variational inequality. The general a

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V. Bostan, W. Han

posteriori error estimate features the presence of a dual variable. Different a posteriori error estimates can be obtained with different choices of the dual variable. In Section 3.4, we make a particular choice of the dual variable that leads to a residual-based error estimate of the finite element solution of the model elliptic variational inequality, and explore the efficiency of the error estimate. In Section 3.5, we make another choice of the dual variable and obtain a recovery-based error estimate of the finite element solution of the model elliptic variational inequality. We also study the efficiency of the error estimator. In Section 3.6, we present some numerical results to illustrate the effectiveness of the estimates in adaptive solution of the elliptic variational inequalities. In Section 3.7, we extend the discussion to solving a steady-state frictional contact problem. Then we turn to an extension of the discussion in adaptively solving timedependent variational inequalities, taking a model quasistatic variational inequality as an example. We begin with an abstract quasistatic variational inequality, introduced in Section 3.8, which contains as special cases several application problems in contact mechanics and hardening plasticity. A backward Euler discretization is used to approximate the time derivative in the quasistatic variational inequality, leading to a sequence of semidiscretized elliptic variational inequalities. An error estimate of the semidiscrete solution is derived. We then focus on a model quasistatic contact problem and derive a posteriori error estimates for finite element solutions of its semidiscretized approximations in Section 3.9, providing both residual type and recovery type error estimates. Finally, the numerical result showing the effectiveness of the error estimates in the adaptive solution of the model quasistatic contact problem is reported in Section 3.10. We now list some notations used repeatedly in the chapter. Let Ω be a bounded domain in Rd , d ≥ 1, with Lipschitz boundary Γ = ∂Ω. For any open subset ω of Ω with Lipschitz boundary ∂ω, we denote by H m (ω), L2 (ω), and L2 (∂ω) the usual Sobolev and Lebesgue spaces with the standard norms k · km;ω := k · kH m (ω) , k · k0;ω := k · kL2 (ω) , and k · k0;∂ω := k · kL2 (∂ω) . Also, we make use of the standard seminorm | · |m,ω on H m (ω). Throughout this chapter we use the same notation v to denote both v ∈ H 1 (Ω) and its trace γv ∈ L2 (Γ ) on the boundary. We reserve the symbol γ for the element sides.

3.2 Model Elliptic Variational Inequality and Its Finite Element Approximation In this section, we introduce a model elliptic variational inequality of the second kind. We comment that the ideas and techniques presented for a posteriori error analysis in solving the model problem can be extended to other elliptic variational inequalities of the second kind; in particular, in Section


31

3.7 we provide a posteriori error analysis for the finite element solution of a steady-state frictional contact problem. Let Ω be a domain in Rd , d ≥ 1, with a Lipschitz boundary Γ . Let Γ1 ⊂ Γ be a relatively closed subset of Γ , and denote Γ2 = Γ \Γ1 the remaining part of the boundary. We allow the extreme situation with Γ1 = ∅ (i.e., Γ2 = Γ ) or Γ1 = Γ (i.e., Γ2 = ∅). Because the boundary Γ is Lipschitz continuous, the unit outward normal vector ν exists a.e. on Γ . We use ∂/∂ν to denote the outward normal differentiation operator, that exists a.e. on Γ . Assume f ∈ L2 (Ω) and g > 0 are given. Over the space V = HΓ11 (Ω) = {v ∈ H 1 (Ω) : v = 0 a.e. on Γ1 },

(3.4)

we define a bilinear form and two functionals: Z a(u, v) = (∇u · ∇v + u v) dx, Ω Z f v dx, (v) = ZΩ g |v| ds. j(v) = Γ2

In the space V , we use the H 1 (Ω)-norm. The model problem is the following elliptic variational inequality of the second kind u ∈ V,

a(u, v − u) + j(v) − j(u) ≥ (v − u)

∀ v ∈ V.

(3.5)

This model is a so-called simplified friction problem following [34], as it can be viewed as a simplified version of a frictional contact problem in linearized elasticity (cf. Section 3.7). The bilinear form a(·, ·) is continuous and V -elliptic, the linear functional (·) is continuous, and the functional j(·) is proper, convex, and continuous, and therefore, by a standard existence and uniqueness result for elliptic variational inequalities of the second kind (see [34, 35]), the variational inequality (3.5) has a unique solution. Moreover, due to the symmetry of the bilinear form a(·, ·), the variational inequality (3.5) is equivalent to the minimization problem: find u ∈ V such that J(u) = inf J(v), v∈V

(3.6)

where J is the energy functional: J(v) =

1 a(v, v) + j(v) − (v). 2

(3.7)

The minimization problem (3.6) also has a unique solution. In the analysis of a posteriori error estimators later, we need the following characterization of the solution u of (3.5): there exists a unique λ ∈ L∞ (Γ2 )

32

V. Bostan, W. Han

such that a(u, v) +

Z

g λ v ds = (v)

Γ2

|λ| ≤ 1,

λ u = |u|

∀ v ∈ V,

a.e. on Γ2 .

(3.8) (3.9)

The function λ can be viewed as a Lagrange multiplier. A proof of this characterization in the case Γ2 = Γ can be found in [34]. The argument there can be extended straightforwardly to the more general situation considered here; see also the proof of Theorem 3.3. It follows from (3.8) that the solution u of (3.5) is the weak solution of the boundary value problem −∆u + u = f

in Ω,

u = 0 on Γ1 , ∂u + gλ = 0 on Γ2 . ∂ν We now turn to finite element approximations of the model problem. For simplicity, we suppose that Ω has a polyhedral boundary Γ . In order to define the finite element method for (3.5) we introduce a family of finite element spaces Vh ⊂ V , which consist of continuous piecewise polynomials of certain degree, corresponding to partitions Ph of Ω into triangular or tetrahedral elements (other kinds of elements, such as quadrilateral elements, or hexahedral or pentahedral elements, can be considered as well). The partitions Ph are compatible with the decomposition of Γ into Γ1 and Γ2 . In other words, if an element side lies on the boundary, then it belongs to one of the sets Γ1 or Γ2 . For every element K ∈ Ph , let hK be the diameter of K and ρK be the diameter of the largest ball inscribed in K. For a side γ of the element K, we denote by hγ the diameter of γ. We assume that the family of partitions Ph , h > 0, satisfies the shape regularity assumption; that is, the ratio hK /ρK is uniformly bounded over the whole family by a constant C. Note that the shape regularity assumption does not require that the elements be of comparable size and thus locally refined meshes are allowed. We use Eh for the set of the element sides; Eh,Γ , Eh,Γ1 , and Eh,Γ2 for the subsets of the element sides lying on Γ , Γ1 , and Γ2 , respectively; and Eh,0 = Eh \Eh,Γ for the subset of the element sides that do not lie on Γ . Let Nh be the set of all nodes in Ph and Nh,0 ⊂ Nh the set of free nodes; that is, those nodes that do not lie on Γ1 . For a given element K ∈ Ph , N (K) and E(K) denote the sets of the nodes of K and sides of K, respectively. e associated with any element K from a partition Ph consists The patch K e = S{K 0 ∈ Ph : of all elements sharing at least one vertex with K; that is, K K 0 ∩ K 6= ∅}. Similarly, for any side γ ∈ Eh , the patch γ e consists of the elements sharing γ as a common side. Note that in the case where the side γ lies on the boundary Γ , the patch γ e consists of only one element. For a given


33

element K ∈ Ph , ν K denotes the unit outward normal vector to the sides of K. When a side γ lies on the boundary Γ , ν γ denotes the unit outward normal vector to Γ . For a side γ in the interior, ν γ is taken to be one of the two unit normal vectors. In what follows, for any piecewise continuous function ϕ and any interior side γ ∈ Eh,0 , [ϕ]γ denotes the jump of ϕ across γ in the direction ν γ ; that is, [ϕ]γ (x) = lim (ϕ(x + t ν γ ) − ϕ(x − t ν γ )) t→0+

∀ x ∈ γ.

In the derivation of a posteriori error estimates we use the so-called weighted Clément-type interpolation operator. There are several variants (see, e.g., [8, 21, 23, 24, 71]) of the interpolation operator introduced by Clément [28]. The main difference among these interpolants lies in the way the interpolation is performed near the boundary. In this chapter we follow the approach used in [23]. Corresponding to the partition Ph , we denote Nv ⊂ Nh to be the set of the element vertices, Nv,Γ1 ⊂ Nv the subset of the element vertices lying on Γ1 , and Nv,0 = Nv ∩ Nh,0 the subset of the interior vertices. Given a ∈ Nv , let ϕa be the linear element nodal basis function associated with a. For each fixed vertex a ∈ Nv,Γ1 , choose ξ(a) ∈ Nv,0 to be an interior vertex of an element containing a. Let ξ(a) = a if a ∈ Nv,0 . For each node a ∈ Nv,0 define the class I(a) = {˜ a ∈ Nv : ξ(˜ a) = a}. In this way, the set of all the vertices Nv is partitioned into card(Nv,0 ) classes of equivalence. For each a ∈ Nv,0 set X ψa = ϕa˜ . ˜ ∈I(a ) a

e a = supp(ψa ) and Notice that {ψa : a ∈ Nv,0 } is a partition of unity. Let K e e ea ha = diam(Ka ). The set Ka is connected and ψa 6= ϕa implies that Γ1 ∩ K has a positive surface measure. For a given v ∈ L1 (Ω), let R e v ψa dx va = RKa (3.10) , a ∈ Nv,0 . e a ϕa dx K

Then define the interpolation operator Πh : V → Vh as follows: X Πh v = va ϕa .

(3.11)

a ∈Nv,0

The next result summarizes some basic estimates for Πh . Its proof can be found in [21]. Theorem 3.2. There exists an h-independent constant C > 0 such that for all v ∈ V and f ∈ L2 (Ω),

34

V. Bostan, W. Han

Z X

γ∈Eh

f (v − Πh v) dx ≤ C|v|1;Ω ⎝

X

a ∈Nv,0

(3.12) ⎞1/2

h2a min kf − fa k20;Ke ⎠ a fa ∈R

,

(3.13)

2 2 kh−1 K (v − Πh v)k0;K ≤ C|v|1;Ω ,

(3.14)

kh−1/2 (v − Πh v)k20;γ ≤ C|v|21;Ω . γ

(3.15)

K∈Ph

X

Ω

|v − Πh v|21;Ω ≤ C|v|21;Ω , ⎛

The finite element method for the variational inequality (3.5) is uh ∈ Vh ,

a(uh , vh −uh )+j(vh )−j(uh ) ≥ (vh −uh )

∀ vh ∈ Vh . (3.16)

The discrete problem has a unique solution uh ∈ Vh by the standard existence and uniqueness result on elliptic variational inequalities. We need the following characterization of the finite element solution, similar to that of the solution of the continuous problem. Theorem 3.3. The unique solution uh ∈ Vh of the discrete problem (3.16) is characterized by the existence of λh ∈ L∞ (Γ2 ) such that Z a(uh , vh ) + g λh vh ds = (vh ) ∀ vh ∈ Vh , (3.17) Γ2

|λh | ≤ 1,

λh uh = |uh |

a.e. on Γ2 .

(3.18)

Proof. Assuming (3.16), let us prove (3.17) and (3.18). Taking first vh = 0 and then vh = 2uh in (3.16), we obtain Z a(uh , uh ) + g |uh | ds = (uh ), (3.19) Γ2

Together with (3.19) the relation (3.16) leads to Z | (vh ) − a(uh , vh )| ≤ g |vh | ds Γ2

∀ vh ∈ Vh .

(3.20)

Write Vh = Vh0 ⊕ Vh⊥ , where Vh0 = Vh ∩ H01 (Ω) and Vh⊥ is the orthogonal complement of Vh in HΓ11 (Ω). It follows from (3.20) that (vh )−a(uh , vh ) = 0 ∀ vh ∈ Vh0 . Notice that the trace operator from Vh⊥ onto Vh⊥ |Γ2 ⊂ L1 (Γ2 ) is an isomorphism. Therefore, the mapping L(vh ) ≡ (e vh ) − a(uh , veh ) can be viewed as a linear functional on Vh⊥ |Γ2 , where veh is any element from the space Vh whose trace on Γ2 is vh . It follows from (3.20) that Z |L(vh )| ≤ g |vh | ds ∀ vh ∈ Vh⊥ |Γ2 . (3.21) Γ2


35

Thus, by the Hahn—Banach theorem the functional L(vh ) can be extended to L(v) on L1 (Γ2 ) and so there exists λh ∈ L∞ (Γ2 ) such that Z L(v) = λh g v ds ∀ v ∈ L1 (Γ2 ) Γ2

and |λh | ≤ 1 a.e. on Γ2 , from which (3.17) follows. Taking now vh = uh in relation (3.17), we have Z a(uh , uh ) + g λh uh ds = (uh ), Γ2

and using (3.19) we get

Z

Γ2

g (|uh | − λh uh ) ds = 0.

Because |λh | ≤ 1 a.e. on Γ2 , we must have |uh | = λh uh a.e. on Γ2 . This completes the proof of (3.17) and (3.18). Conversely, assume (3.17) and (3.18) hold. It follows from relation (3.17) that Z a(uh , vh − uh ) + g λh (vh − uh ) ds = (vh − uh ) ∀ vh ∈ Vh , Γ2

which can be rewritten as Z Z a(uh , vh − uh ) + g λh vh ds − Γ2

Γ2

Then, relation (3.18) implies that Z Z a(uh , vh − uh ) + g λh vh ds − Γ2

g λh uh ds = (vh − uh )

∀ vh ∈ Vh .

g |uh | ds = (vh − uh )

∀ vh ∈ Vh .

Γ2

Because λh vh ≤ |vh | a.e. on Γ2 , it follows immediately that uh is the solution of the discrete problem (3.16). t u Convergence and a priori error estimates for the finite element method (3.16) can be found in the literature (e.g., [34, 35]). Here, we focus on the derivation and analysis of a posteriori error estimators that can be used in the adaptive finite element solution of variational inequalities. In investigation of the efficiency of the a posteriori error estimators, we follow Verf¨ urth [70], with special attention paid to the inequality feature of the problem. The argument makes use of the canonical bubble functions constructed for each element K ∈ Ph and each side γ ∈ Eh . Denote by PK a polynomial space associated with the element K. The following two theorems provide some basic properties of the bubble functions used to derive lower bounds. For more details on bubble functions and proofs see [1].

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V. Bostan, W. Han

Theorem 3.4. Let K ∈ Ph and ψK be its corresponding bubble function. Then there exists a constant C, independent of hK , such that for any v ∈ PK , Z C −1 kvk20;K ≤ ψK v 2 dx ≤ Ckvk20;K , K

C −1 kvk0;K ≤ kψK vk0;K + hK |ψK v|1;K ≤ Ckvk0;K . Theorem 3.5. Let K ∈ Ph and γ ∈ E(K) be one of its sides. Let ψγ be the side bubble function corresponding to γ. Then there exists a constant C, independent of hK , such that for any v ∈ PK , Z C −1 kvk20;γ ≤ ψγ v 2 ds ≤ Ckvk20;γ , γ

−1/2

hK

1/2

kψγ vk0;K + hK |ψγ v|1;K ≤ Ckvk0;γ .

3.3 Dual Formulation and A Posteriori Error Estimation We now present a dual formulation for the model elliptic variational inequality within the framework of Theorem 3.1. The dual formulation is used in the derivation of a posteriori error estimators for approximate solutions. Let us choose the spaces, functionals, and operators needed in applying Theorem 3.1. The space V is defined in (3.4). Let Q = (L2 (Ω))d × L2 (Ω) × L2 (Γ2 ). Any element q ∈ Q is written as q = (q 1 , q2 , q3 ), where q 1 ∈ (L2 (Ω))d , q2 ∈ L2 (Ω), and q3 ∈ L2 (Γ2 ). Let V ∗ and Q∗ = (L2 (Ω))d × L2 (Ω) × L2 (Γ2 ) be the duals of V and Q placed in duality by the pairings h·, ·iV and h·, ·iQ , respectively. Introduce a linear bounded operator Λ : V → Q by the relation Λv = (∇v, v, v|Γ2 )

∀ v ∈ V.

Define F (v, q) =

Z ∙ Ω

¸ Z 1 g |q3 | ds, (|q 1 |2 + |q2 |2 ) − f v dx + 2 Γ2

v ∈ V, q ∈ Q.

Then J(v) = F (v, Λv)

∀ v ∈ V,

and we rewrite the minimization problem (3.6) as u ∈ V,

F (u, Λu) = inf F (v, Λv). v∈V

To apply Theorem 3.1, we need the conjugate function

(3.22)


37

F ∗ (Λ∗ q ∗ , −q ∗ ) ≡ sup {hΛ∗ q ∗ , viV − hq ∗ , qiQ − F (v, q)} , v∈V q ∈Q

where Λ∗ : Q∗ → V ∗ is the adjoint of Λ. We have ½Z Z F ∗ (Λ∗ q ∗ , −q ∗ ) = sup [q ∗1 · ∇v + (q2∗ + f ) v] dx + v∈V q∈Q

Ω

q3∗ v ds

Γ2

Z µ

¶ ¶ Z µ 1 1 2 ∗ 2 ∗ − |q | + q 1 · q 1 dx − |q2 | + q2 q2 dx 2 1 2 Ω Ω ¾ Z − (q3∗ q3 + g |q3 |) ds . (3.23) Γ2

It can easily be verified that ½ Z µ ¶ ¾ Z 1 1 ∗2 2 ∗ − sup |q 1 | + q 1 · q 1 dx = |q 1 | dx, 2 2 d q 1 ∈(L (Ω)) Ω Ω 2 ½ Z µ ¶ ¾ Z 1 1 ∗2 sup − |q2 |2 + q2∗ q2 dx = |q2 | dx, 2 q2 ∈L2 (Ω) Ω Ω 2 ¾ ½ ½ Z 0, if |q3∗ | ≤ g a.e. on Γ2 , ∗ sup (q3 q3 + g |q3 |) ds = − ∞, otherwise. q3 ∈L2 (Γ2 ) Γ2 Note that the term ½Z Z sup [q ∗1 · ∇v + (q2∗ + f ) v] dx + v∈V

Ω

Γ2

equals ∞, unless q ∗ ∈ Q∗ satisfies Z Z [q ∗1 · ∇v + (q2∗ + f ) v] dx + Ω

Γ2

q3∗ v ds = 0

¾ q3∗ v ds

∀v ∈ V ;

and under this assumption, the above term equals 0. Thus, the conjugate function (3.23) is ½R 1 (|q ∗1 |2 + |q2∗ |2 ) dx, if q ∗ ∈ Q∗f,g , Ω 2 (3.24) F ∗ (Λ∗ q ∗ , −q ∗ ) = +∞, otherwise, where the admissible dual function set ½ Z Z Q∗f,g = q ∗ ∈ Q∗ : [q ∗1 · ∇v + (q2∗ + f ) v] dx + q3∗ v ds = 0 Ω Γ2 ¾ ∗ |q3 | ≤ g a.e. on Γ2 . Note that the classical form on the constraint q ∗ ∈ Q∗f,g is

∀ v ∈ V, (3.25)

38

V. Bostan, W. Han

div q ∗1 − q2∗ = f q ∗1 · ν + q3∗ = 0 |q3∗ | ≤ g

in Ω, on Γ2 , on Γ2 .

In conclusion, the dual problem of (3.22) is ª © p∗ ∈ Q∗f,g , −F ∗ (Λ∗ p∗ , −p∗ ) = sup − F ∗ (Λ∗ q ∗ , −q ∗ ) .

(3.26)

q ∗ ∈Q∗ f,g

Existence of solutions of the problems (3.22) and (3.26) is assured by Theorem 3.1, and moreover, F (u, Λu) = −F ∗ (Λ∗ p∗ , −p∗ ). (3.27)

The function q 7→ F ∗ (Λ∗ q ∗ , −q ∗ ) is strictly convex over Q∗f,g , thus a solution of the dual problem (3.26) is unique. Now let w ∈ V be an (arbitrary) approximation of u ∈ V , the unique solution of (3.5). In the rest of the section, we present a general framework for a posteriori estimates of the error u−w. The error bounds are computable from the (known) approximant w. Later on, w is taken as the finite element solution of the variational inequality. Consider 1 1 1 a(u − w, u − w) = a(w, w) − a(u, w) + a(u, u). 2 2 2 By using (3.5) and (3.7), 1 1 1 a(u − w, u − w) = a(w, w) − a(u, w − u) − a(u, u) 2 2 2 1 1 ≤ a(w, w) + j(w) − j(u) − (w − u) − a(u, u) 2 2 = J(w) − J(u). Relation (3.27) implies J(u) = F (u, Λu) = −F ∗ (Λ∗ p∗ , −p∗ ) ≥ −F ∗ (Λ∗ q ∗ , −q ∗ )

∀ q ∗ ∈ Q∗f,g .

Therefore, 1 a(u − w, u − w) ≤ J(w) + F ∗ (Λ∗ q ∗ , −q ∗ ) 2

∀ q ∗ = (q ∗1 , q2∗ , q3∗ ) ∈ Q∗f,g .

Introduce a function space Q∗r = (L2 (Ω))d × L2 (Ω), and write r∗ = (r∗1 , r2∗ ) for any r∗ ∈ Q∗r . We have

(3.28)


J(w) + F ∗ (Λ∗ q ∗ , −q ∗ ) =

Z

Ω

− +

Z

1 (|∇w + r∗1 |2 + |w + r2∗ |2 ) dx 2

Ω

Z

Ω

(∇w · r ∗1 + wr2∗ + f w) dx +

Ω

Z

Γ2

g |w| ds

1 ∗2 (|q | − |r∗1 |2 + |q2∗ |2 − |r2∗ |2 ) dx. 2 1

Because q ∗ = (q ∗1 , q2∗ , q3∗ ) ∈ Q∗f,g , from (3.25), Z

39

(q ∗1 · ∇w + q2∗ w) dx +

Z

Γ2

q3∗ w ds = −

Z

(3.29)

f w dx.

Ω

Using this relation in (3.29) and recalling (3.28), we find that 1 a(u − w, u − w) 2 Z Z 1 1 ∗ ∗ 2 ∗ 2 ≤ (|∇w + r 1 | + |w + r2 | ) dx + (|q 1 − r∗1 |2 + |q2∗ − r2∗ |2 ) dx Ω 2 Ω 2 Z [(q ∗1 − r∗1 ) · (∇w + r∗1 ) + (q2∗ − r2∗ )(w + r2∗ )] dx + ZΩ (g |w| + q3∗ w) ds. + Γ2

So for any q ∗ ∈ Q∗f,g and r∗ ∈ Q∗r , 1 a(u − w, u − w) ≤ 2

Z

(|∇w + r∗1 |2 + |w + r2∗ |2 ) dx Ω Z Z (g |w| + q3∗ w) ds. + (|q ∗1 − r∗1 |2 + |q2∗ − r2∗ |2 ) dx + Ω

Γ2

Thus, we have established the following result. Theorem 3.6. Let u ∈ V be the unique solution of (3.5), and w ∈ V an approximation of u. Then the following estimate holds for any r∗ ∈ Q∗r : Z 1 (|∇w + r∗1 |2 + |w + r2∗ |2 ) dx a(u − w, u − w) ≤ 2 Ω ½Z (|q ∗1 − r∗1 |2 + |q2∗ − r2∗ |2 ) dx + ∗ inf∗ q ∈Qf,g

+

Z

Γ2

Ω

(g |w| +

¾

q3∗ w) ds

.

(3.30)

Consider the second term ½Z ¾ Z ∗ ∗ 2 ∗ ∗ 2 ∗ II ≡ ∗ inf∗ (|q 1 − r1 | + |q2 − r2 | ) dx + (g |w| + q3 w) ds q ∈Qf,g

Ω

Γ2

40

V. Bostan, W. Han

on the right side of the estimate (3.30). From the definition (3.25), ½Z Z II = ∗inf ∗ sup (|q ∗1 − r∗1 |2 + |q2∗ − r2∗ |2 ) dx + (g |w| + q3∗ w) ds q ∈Q v∈V |q3∗ |≤g

Ω

Γ2

+

Z

Ω

[q ∗1 · ∇v + (q2∗ + f ) v] dx +

Z

Γ2

¾ q3∗ v ds .

Here and below, the condition “|q3∗ | ≤ g” stands for “|q3∗ | ≤ g a.e. on Γ2 .” Substitute q ∗1 − r∗1 by q ∗1 , q2∗ − r2∗ by q2∗ , and regroup the terms to get ½Z II = ∗inf ∗ sup (|q ∗1 |2 + q ∗1 · ∇v + |q2∗ |2 + q2∗ v) dx q ∈Q v∈V Ω ∗ Z |q3 |≤g + [r∗1 · ∇v + (r2∗ + f ) v] dx ¾ ZΩ + [g |w| + q3∗ (w + v)] ds Γ2 ¸ ½Z ∙ 1 2 2 ∗ ∗ = inf sup + v ) + r · ∇v + (r + f )v dx − (|∇v| 1 2 4 |q3∗ |≤g v∈V Ω ¾ Z Z ∗ ∗ + q3 v ds + (g |w| + q3 w) ds . Γ2

Γ2

Define the residual R(q3∗ , r∗ )

1 = sup v∈V kvkV

½Z

[r∗1

Ω

· ∇v +

(r2∗

+ f ) v] dx +

Z

¾

q3∗ v ds Γ2

. (3.31)

Then ¾ ½ Z 1 2 ∗ ∗ ∗ II ≤ inf sup + R(q , r ) kvk + (g |w| + q w) ds − kvk V V 3 3 4 |q3∗ |≤g v∈V Γ ¾ 2 ½ Z = inf (g |w| + q3∗ w) ds . R(q3∗ , r∗ )2 + ∗ |q3 |≤g

Γ2

This last estimate is combined with Theorem 3.6, leading to the next result. Theorem 3.7. Let u ∈ V be the unique solution of (3.5), and w ∈ V an approximation of u. Then for any r∗ = (r∗1 , r2∗ ) ∈ Q∗r , Z 1 (|∇w + r∗1 |2 + |w + r2∗ |2 ) dx a(u − w, u − w) ≤ 2 Ω ¾ ½ Z ∗ ∗ 2 ∗ + inf , r ) + (g |w| + q w) ds , (3.32) R(q 3 3 ∗ |q3 |≤g

Γ2

where the residual R(q3∗ , r∗ ) is defined by (3.31).


41

In the next two sections, we make particular choices of r∗1 and r2∗ in (3.32) to obtain residual-based and recovery-based error estimates of finite element solutions of the model elliptic variational inequality.

3.4 Residual-Based Error Estimates for the Model Elliptic Variational Inequality In (3.32), we let r∗1 = −∇w and r2∗ = −w. Then r 1 a(u − w, u − w) ≤ R, 2

(3.33)

where R = inf ∗

|q3 |≤g

(∙

¶¸2 µZ Z 1 ∗ [−∇w · ∇v + (−w + f )v] dx + q3 v ds sup v∈V kvkV Ω Γ2 ¾1/2 Z + (g |w| + q3∗ w) ds . (3.34) Γ2

Although it is possible to derive (3.33)—(3.34) through other approaches, we comment that Theorem 3.7 provides a general framework for various a posteriori error estimates with different choices of the auxiliary variable r∗ . In the limiting case g = 0, the problem (3.5) reduces to the variational equation u ∈ V, a(u, v) = (v) ∀ v ∈ V. Correspondingly, the estimate (3.33)—(3.34) reduces to the familiar form r Z 1 1 [(f − w)v − ∇w · ∇v] dx, a(u − w, u − w) ≤ sup 2 v∈V kvkV Ω which is a starting point in deriving some a posteriori error estimators for Galerkin approximations of linear elliptic partial differential equations (cf. [1]). Now we focus on a posteriori analysis for the finite element solution error, that is, for the situation where the approximant w = uh is the finite element solution. By taking q3∗ = −g λh , and substituting v by vh − v in (3.34) we obtain ½Z 1 R ≤ sup [∇uh · ∇(v − vh ) + (uh − f ) (v − vh )] dx v∈V kvkV Ω ¾ Z + g λh (v − vh ) ds , (3.35) Γ2

42

V. Bostan, W. Han

for any vh ∈ Vh . Here we have used (3.17). For a given v ∈ V , take vh = Πh v in (3.35), where Πh v is defined in (3.11). ½Z 1 R ≤ sup [∇uh · ∇(v − Πh v) + (uh − f ) (v − Πh v)] dx v∈V kvkV Ω ¾ Z g λh (v − Πh v) ds . + Γ2

Decompose the integrals into local contributions from each element K ∈ Ph and integrate by parts over K to obtain X ½Z 1 ∂uh (v − Πh v) ds R ≤ sup ∂ν K v∈V kvkV ∂K K∈Ph Z (−∆uh + uh − f )(v − Πh v) dx + K ¾ Z + g λh (v − Πh v) ds . (3.36) ∂K∩Γ2

Define interior residuals for each element K ∈ Ph by rK = −∆uh + uh − f

in K,

and side residuals for each side γ ∈ Eh,0,Γ2 ≡ Eh,0 ∪ Eh,Γ2 by ( if γ ∈ Eh,0 , [ ∂uh ]γ Rγ = ∂u∂νh if γ ∈ Eh,Γ2 , ∂ν + g λh where the quantity ∙

∂uh ∂ν

¸

γ

(3.37)

(3.38)

= ν K · (∇uh )K + ν K 0 · (∇uh )K 0

represents the jump discontinuity in the approximation to the normal derivative on the side γ which separates the neighboring elements K and K 0 . By using definitions (3.37) and (3.38), relation (3.36) reduces to ⎧ Z 1 ⎨ X R ≤ sup rK (v − Πh v) dx v∈V kvkV ⎩ K K∈Ph

+

X

γ∈Eh,0,Γ2

⎫ ⎬ Rγ (v − Πh v) ds . (3.39) ⎭ γ

Z

Using the estimates (3.14) and (3.15) in (3.39), and applying the Cauchy— Schwarz inequality, we have


⎧ ³ X 1 ⎨ C |v|1;Ω R ≤ sup h2K krK k20;K + v∈V kvkV ⎩ K∈Ph

⎛

≤C⎝

X

K∈Ph

h2K krK k20;K +

X

γ∈Eh,0,Γ2

X

γ∈Eh,0,Γ2

⎞1/2

hγ kRγ k20;γ ⎠

43

hγ kRγ k20;γ

.

⎫ ´1/2⎬ ⎭

(3.40)

We summarize the above results in the form of a theorem. Theorem 3.8. Let u and uh be the unique solutions of (3.5) and (3.16), respectively. Then the error eh = u − uh satisfies the a posteriori estimate ⎛ ⎞ X X h2K krK k20;K + hγ kRγ k20;γ ⎠ , (3.41) keh k2V ≤ C ⎝ K∈Ph

γ∈Eh,0,Γ2

where rK and Rγ are interior and side residuals, defined by (3.37) and (3.38), respectively. In practical computations, the terms on the right side of (3.41) are regrouped by writing X 2 2 2 keh k2V ≤ C ηR , ηR = ηR,K , (3.42) K∈Ph

where the local error indicator ηR,K on each element K, defined by 1 2 ηR,K = h2K krK k20;K + hK 2

X

γ∈E(K)∩Eh,0

kRγ k20;γ + hK

X

γ∈E(K)∩Eh,Γ2

kRγ k20;γ ,

(3.43) identifies contributions from each of the elements to the global error. In the last part of the section, we explore the efficiency of the error bound from (3.41) or (3.42). We derive an upper bound for the error estimator. Integrating by parts over each element and using (3.8) and Theorem 3.18 we have, for any v ∈ V , a(eh , v) = a(u, v) − a(uh , v) Z Z Z f v dx − g λ v ds − (∇uh · ∇v + uh v) dx = Ω Γ2 Ω ∙ ¸ Z X X Z ∂uh = (∆uh − uh + f )v dx + − v ds ∂ν γ K∈Ph K γ∈Eh,0 γ ¶ X Z µ ∂uh + − − gλ v ds. ∂ν γ γ∈Eh,Γ2

Thus,

44

V. Bostan, W. Han

a(eh , v) = −

X Z

K∈Ph

K

rK v dx −

X

γ∈Eh,0,Γ2

Z

Rγ v ds +

γ

X Z

γ∈Eh,Γ2

γ

g(λh − λ)v ds,

(3.44) where rK and Rγ are interior and side residuals defined for each element K ∈ Ph and each side γ ∈ Eh,0,Γ2 by (3.37) and (3.38), respectively. In order to simplify notation, we omit the subscripts K and γ. In the following, we apply Theorems 3.4 and 3.5 choosing PK as the space of polynomials of degree less than or equal to l, and l is any integer larger than or equal to the local polynomial degree of the finite element functions. Let r be a discontinuous piecewise polynomial approximation to the residual r; that is, r|K ∈ PK . Applying Theorem 3.4, we get Z 2 krk0;K ≤ C ψK r2 dx. (3.45) K

Because the function v = ψK r vanishes on the boundary ∂K, it can be extended to a function in V by 0 to the rest of the domain Ω. Inserting this extended function v in the residual equation (3.44), one obtains Z a(eh , ψK r) = − rψK r dx. K

Using this relation, we obtain Z Z 2 ψK r dx = ψK r(r − r) dx − a(eh , ψK r). K

(3.46)

K

The terms on the right side of (3.46) are bounded by making use of the Cauchy—Schwarz inequality and the second part of Theorem 3.4, Z ψK r2 dx ≤ Ckrk0;K kr − rk0;K + C h−1 K keh k1;K krk0;K . K

Combined with (3.45) we have ¡ ¢ krk0;K ≤ C kr − rk0;K + h−1 K keh k1;K .

With the aid of the triangle inequality, finally we get ¡ ¢ krk0;K ≤ kr − rk0;K + krk0;K ≤ C kr − rk0;K + h−1 K keh k1;K .

(3.47)

Consider now an interior side γ ∈ Eh,0 . From the first part of Theorem 3.5 it follows that Z 2 kRk0;γ ≤ C ψγ R2 ds. (3.48) γ

Let γ e denote the subdomain of Ω consisting of the side γ and the two neighγ and as before it can bouring elements. The function v = ψγ R vanishes on ∂e


45

be extended continuously to the whole domain Ω by 0 outside γ e. With this choice of v the residual equation (3.44) reduces to Z Z a(eh , ψγ R) = − rψγ R dx − ψγ R2 ds. Therefore,

Z

γ

γ e

γ

ψγ R2 ds = −a(eh , ψγ R) −

Z

rψγ R dx.

(3.49)

γ e

Applying the Cauchy—Schwarz inequality and the second part of Theorem 3.5 to the terms on the right side of (3.49), we obtain Z ψγ R2 ds ≤ C h−1/2 keh k1;eγ kRk0;γ + C h1/2 γ kRk0;γ , γ γ krk0;e γ

which, combined with (3.48) and (3.47), implies that for every interior side γ ∈ Eh,0 , ³ ´ kRk0;γ ≤ C h−1/2 keh k1;eγ + h1/2 (3.50) γ . γ γ kr − rk0;e Finally, consider those sides γ lying on Γ2 . Denote R ∈ PK an approximation to the residual R = (∂uh /∂ν) + gλh on γ, γ ∈ E(K). The first part of Theorem 3.5 implies Z kRk20;γ ≤ C

2

ψγ R ds.

(3.51)

γ

e be the element whose boundary Define the function v = ψγ R and let γ contains the side γ. Then v|∂eγ \γ = 0. Extend this function to the whole domain by zero value outside γ e. The residual equation (3.44), with this choice of v, becomes Z Z Z a(eh , ψγ R) = − rψγ R dx − Rψγ R ds + g(λh − λ)ψγ R ds, γ e

γ

γ

which leads to Z Z Z Z 2 ψγ R ds = ψγ R(R−R) ds−a(eh , ψγ R)− rψγ R dx+ g(λh −λ)ψγ R ds. γ

γ

γ e

γ

(3.52) As before, the first three terms on the right side of (3.52) can be bounded by applying Theorem 3.5 and the Cauchy—Schwarz inequality. Using (3.51), we then obtain for each side γ ∈ Eh,Γ2 , ³ ´ kRk0;γ keh k1;eγ + h1/2 kRk20;γ ≤ C kRk0;γ kR − Rk0;γ + h−1/2 γ γ γ kRk0;γ krk0;e Z + g(λh − λ)ψγ R ds. γ

46

V. Bostan, W. Han

Multiplying this inequality by hγ and summing over all sides γ ∈ Eh,Γ2 , we get X X 1/2 hγ kRγ k20;γ ≤ C h1/2 γ kRγ k0;γ hγ kRγ − Rγ k0;γ γ∈Eh,Γ2

γ∈Eh,Γ2

+C

X

h1/2 γ γ kRγ k0;γ keh k1;e

γ∈Eh,Γ2

+C

X

h1/2 e k0;e γ + |Rh,Γ2 |, γ kRγ k0;γ hγ krγ

γ∈Eh,Γ2

where

X Z

Rh,Γ2 =

γ∈Eh,Γ2

γ

g (λ − λh ) hγ ψγ Rγ ds.

We can bound Rh,Γ2 as follows, X 1/2 h1/2 |Rh,Γ2 | ≤ γ kg (λ − λh )k0;γ hγ kψγ Rγ k0;γ γ∈Eh,Γ2

⎛

≤C⎝

X

γ∈Eh,Γ2

(3.53)

⎞1/2 ⎛

hγ kλ − λh k20;γ ⎠

⎝

X

γ∈Eh,Γ2

⎞1/2

hγ kRγ k20;γ ⎠

.

Use this bound in (3.53) and apply the Cauchy—Schwarz inequality to get ⎛ X X X hγ kRγ k20;γ ≤ C ⎝ hγ kRγ − Rγ k20;γ + h2γ kreγ k20;eγ γ∈Eh,Γ2

γ∈Eh,Γ2

γ∈Eh,Γ2

+ keh k2V +

X

γ∈Eh,Γ2

⎞

hγ kλ − λh k20;γ ⎠ .

(3.54)

Combining (3.47) and (3.54), we finally conclude that ⎛ X X X hγ kRγ k20;γ ≤ C ⎝ hγ kRγ − Rγ k20;γ + h2γ krγe − reγ k20;eγ γ∈Eh,Γ2

γ∈Eh,Γ2

+ keh k2V +

γ∈Eh,Γ2

X

γ∈Eh,Γ2

Summarizing (3.47), (3.50), and (3.55), we have

⎞

hγ kλ − λh k20;γ ⎠ .

(3.55)


⎛

2 ηR ≤ C ⎝keh k2V +

+

X

K∈Ph

X

γ∈Eh,Γ2

hγ kλ − λh k20;γ

h2K krK − rK k20;K +

X

γ∈Eh,Γ2

47

⎞

hγ kRγ − Rγ k20;γ ⎠.

(3.56)

For our model problem, the element residual rK of (3.37) and the side residual Rγ of (3.38), −∆uh + uh in K and ∂uh /∂ν on γ are polynomials. Therefore, the terms krK −rK k0;K and kRγ −Rγ k0;γ in the right-hand side of (3.56) can be replaced by kf − fK k0;K and kλh − λh,γ k0;γ , with discontinuous piecewise polynomial approximations fK and λh,γ . Theorem 3.9. Let ηR be defined as in (3.42). Then ⎛ X 2 ηR ≤ C ⎝keh k2V + hγ kλ − λh k20;γ +

X

K∈Ph

γ∈Eh,Γ2

h2K kf − fK k20;K +

X

γ∈Eh,Γ2

⎞

hγ kλh − λh,γ k20;γ ⎠

(3.57)

with discontinuous piecewise polynomial approximations fK , λh,γ of f , λh . Let us comment on the three summation terms in (3.57). As P long as2 f has a suitable degree of smoothness, the approximation error K∈Ph hK kf − fK k20;K will be of higher order than keh k2V . Due to the inequality nature of the variational problem, in the efficiency bound (3.57) of the error estimator, P there are extra terms involving λ and λh . A sharp bound of the term γ∈Eh,Γ hγ kλ − λh k20;γ is currently an open problem. Nevertheless, in 2 Section 3.6, we present numerical results showing that the presence of this term in (3.57) does not have an effect on the efficiency of the error estimator. can also be used as an evidence that the term P Similar numerical results 2 h kλ − λ k does not have an effect on the efficiency of the h h,γ 0;γ γ∈Eh,Γ2 γ error estimator.

3.5 Recovery-Based Error Estimates for the Model Elliptic Variational Inequality An important class of a posteriori error estimates is based on local or global averaging of the gradient, for example, in the form of Zienkiewicz—Zhu gradient recovery technique [78, 79, 80]. It is known that in the case of structured grids and higher regularity solutions, such estimators are both efficient and reliable. Some work has been done for unstructured meshes as well (e.g., [74, 72]). In [8, 23], Carstensen and Bartels proved that all averaging tech-

48

V. Bostan, W. Han

niques provide us with a reliable a posteriori error control for the Laplace equation with mixed boundary conditions on unstructured grids also. In the context of solving variational inequalities, gradient recovery type error estimates for elliptic obstacle problem have been derived recently in [9, 71]. In this section, we study a gradient recovery type error estimator for the finite element solution of the model problem, and we restrict our discussion to linear elements. Then all the nodes are also the vertices. To formulate the error estimator we need a gradient recovery operator. There are many types of gradient recovery operators. In order to have a “good” approximation of the true gradient ∇u, a set of conditions to be satisfied by the recovery operator was identified in [1]. These conditions lead to a more precise characterization of the form of the gradient recovery operator, summarized in Lemma 4.5 in [1]. In particular, the recovered gradient at a node a is a linear combination of the values of ∇uh in a patch surrounding a. We define the gradient recovery operator Gh : Vh → (Vh )d as follows: Z X 1 Gh vh (x) = Gh vh (a)ϕa (x), Gh vh (a) = ∇vh dx. (3.58) e a | Ke a |K a ∈Nh Linear elements are used, therefore

Gh vh (a) =

Na X

αia (∇vh )Kai ,

(3.59)

i=1

where (∇vh )Kai denotes the vector value of the gradient ∇vh on the element e a = SNa Kai , αia = |Kai |/|K e a |, i = 1, . . . , Na . Kai , K i=1 Recall from Section 3.4 that residual-type error estimates are derived by applying Theorem 3.7 with r∗ = −(∇uh , uh ), where uh is the finite element solution. In this section, we consider a different choice. Theorem 3.10. Let u and uh be the unique solutions of (3.5) and (3.16), respectively. Then ¶ X µ 2 ku − uh k2V ≤ CηG +C h4a k∇uh k20;Ke + h2a min kf − fa k20;Ke , a

a ∈Nh,0

fa ∈R

a

(3.60)

where 2 ηG =

X

2 ηG,K ,

K∈Ph 2 ηG,K = k∇uh − Gh uh k20;K +

(3.61) X

γ∈E(K)∩Eh,Γ2

hγ kGh uh · ν γ + g λh k20;γ . (3.62)

Proof. Let λh ∈ L∞ (Γ2 ) be provided by Theorem 3.3. Apply Theorem 3.7 with w = uh and r∗ = −(Gh uh , uh ) to obtain


1 a(u − uh , u − uh ) ≤ 2

Z

Ω

|∇uh − Gh uh |2 dx + R2 ,

49

(3.63)

where R = sup v∈V

1 kvkV

½Z

Ω

[Gh uh · ∇v + (uh − f )v] dx +

Z

Γ2

¾ gλh v ds .

(3.64)

Let Πh be the interpolation operator defined by (3.11). Use (3.17) with vh = Πh v: Z Z [∇uh · ∇Πh v + (uh − f ) Πh v] dx + g λh Πh v ds = 0. Ω

Γ2

Therefore, we can rewrite (3.64) as ½Z 1 R = sup [Gh uh · ∇(v − Πh v) + (uh − f )(v − Πh v)] dx v∈V kvkV Ω ¾ Z Z gλh (v − Πh v) ds + (Gh uh − ∇uh ) · ∇Πh v dx . + Γ2

Ω

By (3.12), we have k∇Πh vk0;Ω ≤ C kvkV , and so, sup v∈V

1 kvkV

Z

Ω

(Gh uh − ∇uh ) · ∇Πh v dx ≤ C k∇uh − Gh uh k0;Ω .

Thus, R ≤ C k∇uh − Gh uh k0;Ω + R1 ,

(3.65)

where ½Z 1 [Gh uh · ∇(v − Πh v) + (uh − f )(v − Πh v)] dx v∈V kvkV Ω ¾ Z + gλh (v − Πh v) ds .

R1 = sup

Γ2

Integrate by parts over each element K ∈ Ph to get X ½Z 1 R1 = sup (−div(Gh uh ) + uh − f )(v − Πh v) dx v∈V kvkV K K∈Ph X Z + Gh uh · ν γ (v − Πh v) ds γ∈E(K) γ ¾ Z X + g λh (v − Πh v) ds . (3.66) γ∈E(K)∩Eh,Γ2

γ

50

V. Bostan, W. Han

Because Gh uh is continuous, the integrals on the interior sides γ ∈ Eh,0 cancel each other. Write −div(Gh uh ) + uh − f = div(∇uh − Gh uh ) + (−∆uh + uh − f ) and rearrange the terms in (3.66) to obtain ( X Z 1 R1 = sup div(∇uh − Gh uh )(v − Πh v) dx v∈V kvkV K∈Ph K X Z + rK (v − Πh v) dx K∈Ph

+

K

X Z

γ∈Eh,Γ2

= sup v∈V

γ

)

(Gh uh · ν γ + g λh )(v − Πh v) ds ,

ª 1 © I + II + III , kvkV

(3.67)

where rK = −∆uh + uh − f = uh − f denotes the interior residual on element K ∈ Ph . We use r to denote the piecewise interior residual; that is, r|K = rK for K ∈ Ph . To estimate the first summand on the right side of (3.67), we use an elementwise inverse inequality of the form kdiv(∇uh − Gh uh )k0;K ≤ Ch−1 K k∇uh − Gh uh k0;K .

(3.68)

Apply the Cauchy—Schwarz inequality, the inverse inequality (3.68), and the estimate (3.14) to get X I ≤C kdiv(∇uh − Gh uh )k0;K kv − Πh vk0;K K∈Ph

≤C ≤C

X

K∈Ph

Ã

k∇uh − Gh uh k0;K kh−1 K (v − Πh v)k0;K

X

K∈Ph

≤ C |v|1;Ω

k∇uh − Gh uh k20;K

Ã

X

K∈Ph

!1/2 Ã

X

K∈Ph

k∇uh − Gh uh k20;K

!1/2

kh−1 K (v

− Πh v)k20;K

.

(3.69)

For the second summand, we apply the estimate (3.13) to obtain ⎛

II ≤ C|v|1;Ω ⎝

X

a ∈Nh,0

⎞1/2

h2a min kr − ra k20;Ke ⎠ ra ∈R

!1/2

a

.


51

Now X

a ∈Nh,0

h2a min kr − ra k20;Ke ≤ 2 a

ra ∈R

X

a ∈Nh,0

+2

h2a kuh − uh k20;Ke

X

a ∈Nh,0

a

h2a min kf − fa k20;Ke , a fa ∈R

e a . Use the Poincaré inequalwhere uh denotes the integral mean of uh over K ity and an inverse inequality of the form (3.68) to get X X h2a min kr − ra k20;Ke ≤ C h4a k∇uh k20;Ke a ∈Nh,0

a

ra ∈R

a

a ∈Nh,0

+2

X

a ∈Nh,0

h2a min kf − fa k20;Ke . a

fa ∈R

Therefore, II ≤ C|v|1;Ω

¶ X µ 4 2 2 2 ha k∇uh k0;Ke + ha min kf − fa k0;Ke . a

a ∈Nh,0

a

fa ∈R

(3.70)

Finally, with the aid of the Cauchy—Schwarz inequality and the estimate (3.15), the third summand on the right side of (3.67) can be bounded by ⎛

III ≤ C |v|1;Ω ⎝

X

γ∈Eh,Γ2

⎞1/2

hγ kGh uh · ν γ + g λh k20;γ ⎠

.

(3.71)

Inserting (3.69) through (3.71) into (3.67), and using the Cauchy—Schwarz inequality and (3.65), we deduce that ⎧ ⎨ X X R≤C k∇uh − Gh uh k20;K + hγ kGh uh · ν γ + g λh k20;γ ⎩ K∈Ph

γ∈Eh,Γ2

⎫ ¶⎬1/2 X µ + . h4a k∇uh k20;Ke + h2a min kf − fa k20;Ke a a ⎭ f ∈R

(3.72)

a

a ∈Nh,0

Split the first term on the right side of estimate (3.63) into local contributions from each K ∈ Ph and insert (3.72) to conclude the proof. t u We can write (3.60) as ku − uh k1;Ω ≤ CηG + Rh , where

(3.73)

52

V. Bostan, W. Han

⎡

Rh = C ⎣

X

a ∈Nh,0

⎤1/2

h4a k∇uh k20;Ke ⎦ a

⎡

+C⎣

X

a ∈Nh,0

⎤1/2

h2a min kf − fa k20;Ke ⎦ fa ∈R

a

.

We observe that usually the term Rh is of higher order compared to ku − uh k1;Ω which is of order O(h) in the nondegenerate situations. This observation is argued as follows. First, it is easy to show from the definition of the finite element solutions that there is a constant C such that kuh k1;Ω ≤ C for any h. So the term ⎡ ⎤1/2 X ⎣ h4a k∇uh k2 e ⎦ a ∈Nh,0

0;Ka

is bounded by O(h2 ). Next, for f ∈ L2 (Ω), ⎛ ⎞1/2 X ⎝ h2a min kf − fa k20;Ke ⎠ = o(h); a a ∈Nh,0

fa ∈R

and if f ∈ H 1 (Ω), then ⎛ ⎞1/2 X ⎝ h2a min kf − fa k20;Ke ⎠ = O(h2 ). a ∈Nh,0

fa ∈R

a

Thus, (3.73) illustrates the reliability of the error estimator ηG . We now turn to the efficiency of the estimator. We relate the gradient recovery-based estimator ηG to the residual type estimator ηR . Recall that for the residual-type estimator ηR,K is defined in (3.43) with the interior residual rK = −∆uh + uh − f in K and the side residual ½ ∂uh [ ] if γ ∈ Eh,0 , Rγ = ∂u∂νh γ (3.74) if γ ∈ Eh,Γ2 . ∂ν + g λh 2 = For the error estimator ηR its efficiency.

P

K∈Ph

2 ηR,K , we have the inequality (3.57) for

Lemma 3.1. Let ηG,K be defined in (3.62). Then the following bound holds: ⎛ ⎞ X X 2 ≤C⎝ hγ kRγ k20;γ + hγ 0 kRγ 0 k20;γ 0 ⎠ , (3.75) ηG,K γ∈E(K)∩Eh,Γ2

γ 0 ∈EK f

e corresponding to the where EKe denotes the set of inner sides of the patch K element K. Proof. It follows from the definition of Gh that we have on each element K,


¯ X ¯ |∇uh − Gh uh | = ¯¯ 2

ϕa

a ∈N (K)

≤C

X

e K 0 ⊂K

Na ³X i=1

´¯¯2 αa ((∇uh )K − (∇uh )Kai ) ¯¯

53

i

|(∇uh )K − (∇uh )K 0 |2 .

(3.76)

e there is a sequence of inner edges γ1 , . . . , γm , such that For any K 0 ⊂ K, 0 γej ∩ γg f j+1 6= ∅ and K ⊂ γe1 , K ⊂ γ m . Hence, |(∇uh )K − (∇uh )

K0

m X X ¯ ¯ ¯ ¯ ¯[∇uh ]γj ¯ ≤ ¯[∇uh ]γ ¯. |≤ j=1

(3.77)

γ∈EK f

Because uh is continuous on Ω, [∂uh /∂t]γ = 0 for all γ ∈ Eh,0 , where ∂uh /∂t is the tangential derivative of uh . Therefore, |[∇uh ]γ | = |[∂uh /∂ν]γ | if γ ∈ Eh,0 . The estimates (3.76) and (3.77) together with the shape regularity of the partition Ph imply ¸2 Z ∙ X ∙ ∂uh ¸2 X ∂uh 2 2 k∇uh − Gh uh k0;K ≤ ChK ≤C hγ ds. (3.78) ∂ν γ ∂ν γ γ γ∈EK f

γ∈EK f

Let K ∈ Ph be such that E(K) ∩ Eh,Γ2 = ∅. It follows from (3.78) and definitions of ηG,K and Rγ that X 2 ≤C hγ kRγ k20;γ . (3.79) ηG,K γ∈EK f

Consider now the case when the element K has at least one side lying on the boundary Γ2 . Let γ ∈ E(K) ∩ Eh,Γ2 . Apply the triangle inequality to get X 2 = k∇uh − Gh uh k20;K + hγ kGh uh · ν γ + gλh k20;γ ηG,K γ∈E(K)∩Eh,Γ2

≤ k∇uh − Gh uh k20;K X 2 + hγ (k(∇uh − Gh uh ) · ν γ k0;γ + kRγ k0;γ ) γ∈E(K)∩Eh,Γ2

≤ k∇uh − Gh uh k20;K X ¡ ¢ + 2hγ k(∇uh − Gh uh ) · ν γ k20;γ + kRγ k20;γ .

(3.80)

γ∈E(K)∩Eh,Γ2

From an inverse inequality and (3.78), we have hγ k(∇uh − Gh uh ) · ν γ k20;γ ≤ hK k∇uh − Gh uh k20;∂K X ≤ Ck∇uh − Gh uh k20;K ≤ C hγ 0 kRγ 0 k20;γ 0 . (3.81) γ 0 ∈EK f

54

V. Bostan, W. Han

Inserting (3.78) and (3.81) into (3.80) concludes the proof. From Lemma 3.1 and the inequality (3.57) we obtain ⎛ X 2 ηG ≤ C ⎝ku − uh k2V + hγ kλ − λh k20;γ γ∈Eh,Γ2

+

X

K∈Ph

h2K kf − fK k20;K +

X

γ∈Eh,Γ2

t u

⎞

hγ kλh − λh,γ k20;γ ⎠

(3.82)

with discontinuous piecewise polynomial approximations fK and λh,γ . The comments at the end of Section 3.4 apply to the three summation terms in (3.82).

3.6 Numerical Example on the Model Elliptic Variational Inequality In this section, we provide some numerical results on a two-dimensional elliptic variational inequality to illustrate the effectiveness of the error estimators ηR of (3.42)—(3.43) and ηG of (3.61)—(3.62). We use triangular partitioning and linear elements for discretization, and a seven-point Gauss—Legendre quadrature to compute the load vector on each triangle. Numerical integration over a general triangle is done by the reference element technique. On the reference element ˆ = {(ξ, η) : ξ ≥ 0, η ≥ 0, 1 − ξ − η ≥ 0}, K the seven-point Gauss—Legendre quadrature formula is defined by Z

ˆ K

F (ξ, η) dξdη ≈

7 X

ωi F (ξi , ηi ),

i=1

where the nodes {(ξi , ηi )}7i=1 and weights {ωi }7i=1 are given in Table 3.1. The discretized solution is computed by solving the equivalent minimization problem using an overrelaxation method with a relative error tolerance, in the maximum norm, of 10−6 (see [34, 35]). In order to show the effectiveness of the adaptive procedure we compare numerical convergence orders of the approximate solutions. We compute these orders by considering families of uniform and adaptively refined partitions. Consider a sequence of finite element solutions uun h based on uniform partitions of the domain Ω. Starting with an initial coarse partition P1 , we construct a family of nested meshes by subdividing each element into four congruent elements for the two-dimensional case. The solution from the most


55

Table 3.1 Nodes and weights of a quadrature over reference triangle i 1 2 3 4 5 6 7

ξi 1/3 √ (6 + 15)/21 √ (9 − 2 15)/21 √ (6 + 15)/21 √ (6 − 15)/21 √ (9 + 2 15)/21 √ (6 − 15)/21

ηi 1/3 √ (6 + 15)/21 √ (6 + 15)/21 √ (9 − 2 15)/21 √ (6 − 15)/21 √ (6 − 15)/21 √ (9 + 2 15)/21

ωi 9/80 √ (155 + 15)/2400 √ (155 + 15)/2400 √ (155 + 15)/2400 √ (155 − 15)/2400 √ (155 − 15)/2400 √ (155 − 15)/2400

refined mesh is taken as the “true” solution u used to compute the errors of the approximate solutions obtained on the other meshes. Adaptive finite element solutions are obtained by the following algorithm. 1. Start with the initial partition Ph and corresponding finite element subspace Vh . 2. Compute the finite element solution uad h ∈ Vh . 3. For each element K ∈ Ph compute the error estimator ηK , defined in (3.43) for the P residual type and (3.62) for the gradient recovery type. 4. Let η = N1 K∈Ph ηK with N being the number of elements in partition Ph . An element K is marked for refinement if ηK > μ η, where μ is a prescribed threshold. In the example of this section, μ = 0.5. 5. Perform refinement and obtain a new triangulation Ph . 6. Return to step 2. In the computation of the error indicator ηK we make use of the multiplier λh defined on Γ2 ⊂ Γ . In what follows we describe how λh can be (approximately) recovered from the solution uh using characterization (3.17). We compute the piecewise constant and the piecewise linear approximations to the Lagrange multiplier. Denote by {ai }m i=1 the nodes of the partition Ph belonging to Γ2 . Let i m {ϕi }m i=1 be the basis functions corresponding to the nodes {a }. Let {χi }i=1 be the characteristic functions of the intervals {Ki } belonging to Γ2 and defined as follows: Ki is the intersection of Γ2 and the segment joining the midpoints of edges sharing ai as a common point. We first determine a piecewise constant function m X (0) λh,1 = λ0,i h,1 χi i=1

or a piecewise linear function

(1)

λh,1 =

m X i=1

by requiring an analogue of (3.17):

λ1,i h,1 ϕi

56

V. Bostan, W. Han

a(uh , vh ) +

Z

(k)

Γ2

g λh,1 vh ds = (vh )

∀ vh ∈ Vh

(3.83) (k)

with k = 0 (piecewise constant) or k = 1 (piecewise linear). Denote λh,1 = (k)

k,m T (λk,1 h,1 , . . . , λh,1 ) , k = 0, 1. We then project the components of λh,1 onto the (k)

k,m T interval [−1, 1] to get λh,2 = (λk,1 h,2 , . . . , λh,2 ) : k,i λk,i h,2 = max{min{λh,1 , 1}, −1},

i = 1, . . . , m.

(0)

The piecewise constant approximation λh,2 and the piecewise linear approx(1)

imation λh,2 of the multiplier λh on Γ2 ⊂ Γ can be computed as (0)

λh,2 =

m X

λ0,i h,2 χi

and

(1)

λh,2 =

i=1

m X

λ1,i h,2 ϕi .

(3.84)

i=1

(k)

We briefly comment on the method for finding λh,1 , k = 0, 1. Let n = dim Vh . Denote by K the standard (n × n) stiffness matrix and by l ∈ Rn the standard load vector. Let u ∈ Rn be the nodal value vector of the finite element solution uh . Then the algebraic representation of (3.83) becomes (k)

(Ku, v)Rn + (gM λh,1 , v c )Rm = (l, v)Rn

∀ v ∈ Rn ,

(3.85)

where v c denotes the subvector of v, containing the nodal values of vh at the nodes {ai }m i=1 ⊂ Γ2 and M is a sparse (m × m) matrix. We can write v = (v Ti , v Tc )T ∈ Rn−m × Rm by assuming that the components of v c are listed last. We similarly split l to li and lc . This decomposition yields a block structure for K, µ ¶ K ii K ic K= . K ci K cc Then (3.85) is equivalent to the following two relations: K ii ui + K ic uc = li , (k)

K ci ui + K cc uc + gM λh,1 = lc . Once the approximate solution uh is computed, we can obtain from the second relation that (k)

λh,1 = g −1 M −1 (lc − K ci ui − K cc uc ),

k = 0, 1.

ad We use uun h for finite element solutions on uniform meshes, and uh for finite element solutions on adaptive meshes. We find that uses of piecewise linears and piecewise constants for the Lagrange multiplier lead to negligible differences in the adaptive meshes. For instance, in Example 3.1, we get


57

identical adaptive meshes for the first three adaptive steps, and there is a difference of only one or two nodes for the fourth and fifth adaptive steps. Thus, in all the numerical examples in this chapter, uad h refers to the adaptive finite element solution where piecewise linear functions are used for approximating the Lagrange multiplier in generating the adaptive mesh. Because adaptive solutions are involved, numerical solution errors are plotted against the number of degrees of freedom, rather than the meshsize. Example 3.1. Let Ω = [0, 1] × [0, 1] and Γ2 = Γ . We solve the problem of finding u ∈ H 1 (Ω) such that ∀ v ∈ H 1 (Ω), Z Z Z Z [∇u · ∇(v − u) + u (v − u)] dx + g |v| ds − g |u| ds ≥ f (v − u) dx Ω

Γ

Γ

Ω

where f = −∆w + w, w = w1 − w2 , and for i = 1, 2, wi (x) = (i)

½

exp(1/(ri2 − 1)) 0

if ri < 1, otherwise

(i)

with ri = [(x1 −x1,0 )2 +(x2 −x2,0 )2 ]1/2 /εi . For the numerical results reported (1)

(1)

(2)

(2)

below, we let g = 1, x1,0 = 0.8, x2,0 = 0.1, x1,0 = 0.3, x2,0 = 0.1, ε1 = 0.25, and ε2 = 0.2. We start with a coarse uniform triangulation shown on the left plot in Figure 3.2. Here, the interval [0, 1] is divided into 1/h equal parts with h = 1/4 which is successively halved. The numerical solution corresponding to h = 1/256 (66,049 nodes) is taken as the “true” solution u, shown in Figure 3.1. We use the regular refinement technique (red-blue-green refinement), in which the triangle is divided into four triangles by joining the midpoints of edges and adjacent triangles are refined in order to avoid hanging nodes. For a detailed description of this and other refinement techniques currently used see, for example, [70] and references therein. Also, in order to improve the quality of triangulation, a smoothing procedure is used after each refinement. For each triangle K of the triangulation we compute the triangle quality measure defined by √ 4 3 area(K) Q(K) = 2 , h1 + h22 + h23 where hi , i = 1, 2, 3, are the side lengths of the triangle K. Note that Q(K) = 1 if h1 = h2 = h3 . A triangle is viewed to be of acceptable quality if Q > 0.6, otherwise we modify the mesh by moving the interior nodes toward the center

58

V. Bostan, W. Han

0.6 0.4

Solution

0.2 0 −0.2 −0.4 −0.6 −0.8 1 0.8

1 0.6

0.8 0.6

0.4 0.4

0.2

0.2 0

y

0

x

Fig. 3.1 “True” solution.

of mass of the polygon formed by the adjacent triangles. The adaptively refined triangulation after five iterations is shown on the right plot of Figure 3.2. To have an idea of the convergence behaviour of the discrete Lagrange (k) multipliers, we analyze the errors kλ(j) − λh k0;Γ , j, k = 0, 1 corresponding to the sequence of uniform refinements. Here, λ(0) and λ(1) are the piecewise constant and the piecewise linear approximations to the Lagrange multiplier (1) (1) corresponding to the parameter h = 1/256. Graphs of λh,1 and λh,2 with

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.2

0.4

0.6

0.8

1

Fig. 3.2 Initial and adaptively refined partitions.

0

0

0.2

0.4

0.6

0.8

1

3 Finite Element Solution of Variational Inequalities with Applications 1.5

59

0 Side: [0,1]×{0}

Side: {1}×[0,1]

1 0.5

−0.5

0 −0.5

−1

−1 −1.5

0

0.2

0.4

0.6

0.8

1

−1.5

0

0.2

0.4

x

0.6

0.8

1

0.6

0.8

1

y

0

0.3 Side: [0,1]×{1}

Side: {0}×[0,1]

−0.005

0.2

−0.01 0.1 −0.015 0

−0.02 −0.025

0

0.2

0.4

0.6

0.8

1

−0.1

0

0.2

0.4

x

y

(1)

Fig. 3.3 Plots of λh,1 .

(1)

h = 1/256 are provided in Figures 3.3 and 3.4. Note the peaks of λh,1 around x1 = 0.1 and x1 = 0.5 (upper left plot), and x2 = 0.3 (upper right plot), which (1) are eliminated by the projection of λh,1 onto the interval [−1, 1]. Figures 3.5

1.5

0 Side: [0,1]×{0}

Side: {1}×[0,1]

1 0.5

−0.5

0 −0.5

−1

−1 −1.5

0

0.2

0.4

0.6

0.8

1

−1.5

0

0.2

0.4

x

0.6

0.8

1

0.6

0.8

1

y

0

0.3 Side: [0,1]×{1}

Side: {0}×[0,1]

−0.005

0.2

−0.01 0.1 −0.015 0

−0.02 −0.025

0

0.2

0.4

0.6 x

(1)


0.8

1

−0.1

0

0.2

0.4 y

60

V. Bostan, W. Han 1.5

0 Side: [0,1]×{0}

Side: {1}×[0,1]

1 0.5

−0.5

0 −0.5

−1

−1 −1.5

0

0.2

0.4

0.6

0.8

1

−1.5

0

0.2

0.4

x

0.6

0.8

1

0.6

0.8

1

y

0

0.3 Side: [0,1]×{1}

Side: {0}×[0,1]

−0.005

0.2

−0.01 0.1 −0.015 0

−0.02 −0.025

0

0.2

0.4

0.6

0.8

1

−0.1

0

0.2

0.4

x

y

(0)


(0)

(0)

and 3.6 contain the graphs of λh,1 and λh,2 corresponding to h = 1/256. (k)

1/2 kλ(j) − λh k0;Γ , Figure 3.7 provides the error values ku − uun h k1;Ω and h (k) j, k = 0, 1. The numerical convergence orders of h1/2 kλ(j) − λh k0;Γ are

1.5

0 Side: [0,1]×{0}

Side: {1}×[0,1]

1 0.5

−0.5

0 −0.5

−1

−1 −1.5

0

0.2

0.4

0.6

0.8

1

−1.5

0

0.2

0.4

x

0.6

0.8

1

0.6

0.8

1

y

0

0.3 Side: [0,1]×{1}

Side: {0}×[0,1]

−0.005

0.2

−0.01 0.1 −0.015 0

−0.02 −0.025

0

0.2

0.4

0.6 x

(0)


0.8

1

−0.1

0

0.2

0.4 y


61

1

10

solution lambda (j=0, k=0) lambda (j=0, k=1) lambda (j=1, k=0) lambda (j=1, k=1) 0

10

1

Error

0.47

−1

10

1 0.65 −2

10

0.72 1 −3

10

1

2

10

10

3

4

10 Number of degrees of freedom

10

(k)

1/2 kλ(j) − λ Fig. 3.7 Errors ku − uun h k1;Ω (¤ ) versus h h k0;Γ (j, k = 0, 1).

obviously higher than that of ku − uun h k1;Ω , indicating that the second term within the parentheses in the efficiency bounds (3.57) and (3.82) is expected to be of higher order compared to the first term keh k21;Ω . We use an adaptive procedure based on both residual type and gradient recovery type estimates to obtain a sequence of approximate solutions uad h . The adaptive finite element mesh after five adaptive iterations is shown on the right plot in Figure 3.2. Figures 3.8 and 3.9 contain the error values ad ku − uun h k1;Ω and ku − uh k1;Ω . We observe a substantial improvement of the efficiency using P adaptively refined meshes. Figures 3.10 and 3.11 provide 2 the values of ηI = ( K ηI,K )1/2 , I ∈ {R, G}, where ηI,K are computed using either residual type estimator (I = R) or gradient recovery type estimator (I = G) on both uniform and adapted meshes. Table 3.2 contains the values of CI computed for uniform and adaptive solutions: CI =

ku − uh k1;Ω , ηI

I ∈ {R, G}.

Table 3.2 Numerical values of CR and CG h un CR ad CR un CG ad CG

1/4 1.01e-01 1.01e-01 1.52e+00 1.52e+00

1/8 1.36e-01 1.36e-01 1.15e+00 1.05e+00

1/16 1.60e-01 1.60e-01 1.05e+00 1.04e+00

1/32 1.66e-01 1.67e-01 9.50e-01 9.69e-01

1/64 1.64e-01 1.64e-01 9.24e-01 8.30e-01

1/128 1.61e-01 1.65e-01 9.09e-01 8.33e-01

62

V. Bostan, W. Han 1

10

uniform mesh adapted mesh

0

Error

10

1 0.47

−1

10

0.68 1

−2

10

1

10

2

10

3

4


10

Fig. 3.8 Results based on residual type estimator.

It is seen from Table 3.2 that for this numerical example, the gradient recovery type error estimator provides a better prediction of the true error than the residual type error estimator, a phenomenon observed in numerous

1

10


0

Error

10

1 0.47

−1

10

0.68 1

−2

10

1

10

2

10

3


Fig. 3.9 Results based on recovery type estimator.

4

10


63

2

10

uniform mesh error uniform mesh ηR adapted mesh error adapted mesh ηR 1

10

Error

1 0.46 0

10

0.68 1 −1

10

−2

10

1

10

2

10

3

4


10


references. In general, we use the a posteriori error estimates only for the purpose of designing adaptive meshes, due to the presence of the unknown constants CR and CG .

1

10

uniform mesh error uniform mesh ηG adapted mesh error adapted mesh ηG

0

Error

10

1 0.45

−1

10

0.68 1

−2

10

1

10

2

10

3



4

10

64

V. Bostan, W. Han 1

10

adapted mesh (residual) adapted mesh (gradient recovery)

0

Error

10

−1

10

0.68 1

−2

10

1

10

2

10

3

4


10

Fig. 3.12 Performance comparison of the two error estimators.

For comparison of the performance between the two error estimators, we show in Figure 3.12 the errors of the adaptive solutions corresponding to the two error estimators. We observe that the two error estimators lead to very similar solution accuracy for same amount of degrees of freedom. t u More numerical results can be found in [14].

3.7 Application to a Frictional Contact Problem In this section, we take a frictional contact problem as an example to show that similar a posteriori error estimates can be derived for more complicated variational inequalities. Again, we denote by Ω ⊂ Rd (d ≤ 3 in applications) a Lipschitz domain with boundary Γ . The outward unit normal exists a.e. on Γ and is denoted by ν. We use Sd for the space of second-order symmetric tensors on Rd . The canonical inner products and corresponding norms on Rd and Sd are u · v = ui vi , σ : ε = σij εij ,

|v| = (v · v)1/2

|σ| = (σ : σ)1/2

∀ u, v ∈ Rd , ∀ σ, ε ∈ Sd .

Here, the indices i and j run between 1 and d, and the summation convention over repeated indices is used. We define the product spaces L2 (ω) := (L2 (ω))d


65

Pd and H 1 (ω) := (H 1 (ω))d equipped with the norms kvk2k;ω := i=1 kvi k2k;ω , k = 0, 1. When no ambiguity occurs, we use the same notation v to denote the function and its trace on the boundary. For a vector v, we use its normal component vν = v·ν and tangential component v τ = v−vν ν at a point on the boundary. Similarly for a tensor σ ∈ Sd , we define its normal component σν = σν · ν and tangential component σ τ = σν − σν ν. For a detailed treatment of traces for vector and tensor fields in contact problems and related spaces see [53] or [46]. The material occupying Ω is assumed linearly elastic. We denote by C : Ω × Sd → Sd the elasticity tensor of the material. The fourth-order tensor C is assumed to be bounded, symmetric, and positive definite in Ω. We briefly describe the physical setting of the frictional contact problem. Details and other related problems can be found in [53, 46]. Consider an elastic body occupying a bounded domain Ω in Rd , d ≤ 3, with a Lipschitz boundary. The boundary Γ is partitioned as follows: Γ = ΓD ∪ ΓN ∪ ΓC with ΓD , ΓN , and ΓC relatively open and mutually disjoint, and meas(ΓD ) > 0. The subscripts “D”, “N ”, and “C” are intended as shorthand indications for Dirichlet, Neumann, and contact boundary conditions. We assume that the body is clamped on ΓD ; on the boundary part ΓN surface tractions of density f 2 ∈ (L2 (ΓN ))d are applied and on ΓC the body is in bilateral contact with a rigid foundation. The contact is frictional and is modeled by Tresca’s law. Volume forces of density f 1 ∈ (L2 (Ω))d act in Ω. In classical formulation, the problem is to find a displacement field u : Ω → Rd and a stress field σ : Ω → Sd such that σ = Cε(u) ε(u) = 12 (∇u + (∇u)T ) Div σ + f 1 = 0 u=0 σν = f 2 uν = 0 |σ τ | ≤ g |σ τ | < g ⇒ uτ = 0 |σ τ | = g ⇒ uτ = −κ σ τ for some κ ≥ 0

in Ω, in Ω, in Ω, on ΓD , on ΓN , on ΓC , on ΓC , on ΓC , on ΓC ,

(3.86) (3.87) (3.88) (3.89) (3.90) (3.91) (3.92) (3.93) (3.94)

where the friction bound g > 0 on ΓC . We comment that (3.86) is the constitutive relation of the linearized elasticity material, (3.87) defines the linearized strain and the displacement, and (3.88) is the equilibrium equation. The classical displacement and traction boundary conditions are given in (3.89) and (3.90). Contact conditions are described in (3.91)—(3.94). The bilateral contact feature is reflected by (3.91). The relations (3.92)—(3.94) represent Tresca’s friction law.

66

V. Bostan, W. Han

In certain situations, the frictional contact problem stated above describes the material deformation quite accurately. In more complicated situations, such as when the contact zone is not prescribed a priori or when more realistic frictional contact laws are used, the frictional contact problem here can be viewed as an intermediate problem for a typical step in an iterative solution procedure for solving the more complicated contact problem. To introduce a variational formulation of the problem, we let V = {v ∈ H 1 (Ω) : v|ΓD = 0, vν |ΓC = 0} with its inner product and norm defined by Z 1/2 (u, v)V = ε(u) : ε(v) dx, kvkV = (v, v)V . Ω

Because meas(ΓD ) > 0, the Korn inequality holds and we see that kvkV is a norm over V which is equivalent to the canonical norm kvk1;Ω . Over the space V , we define Z a(u, v) = Cε(u) : ε(v) dx, (3.95) Ω Z Z f 1 · v dx + f 2 · v ds, (3.96) (v) = Ω ΓN Z g|v τ | ds. (3.97) j(v) = ΓC

A standard procedure leads to the following variational formulation of the problem (3.86)—(3.94): u∈V,

a(u, v − u) + j(v) − j(u) ≥ (v − u)

∀v ∈ V .

(3.98)

The bilinear form a(·, ·) : V × V → R is obviously continuous. Due to the assumption meas(ΓD ) > 0, it is V -elliptic. The functional : V → R is linear and continuous, and j : V → R is proper and lower semicontinuous convex. Therefore, the variational inequality (3.98) has a unique solution u in V . Moreover, because the bilinear form a(·, ·) is symmetric and positive definite, solving the variational inequality (3.98) is equivalent to minimizing the energy functional J(v) =

1 a(v, v) − (v) + j(v) 2

over the space V . The unique solution u ∈ V of the problem (3.98) is characterized by the existence of a Lagrange multiplier λτ ∈ (L∞ (ΓC ))d such that


a(u, v) +

Z

ΓC

|λτ | ≤ 1,

g λτ · v τ ds = (v)

λτ · uτ = |uτ |

∀v ∈ V , a.e. on ΓC .

67

(3.99) (3.100)

Turn now to a finite element approximation of the problem. Assume Ω has a polyhedral boundary Γ . Let {Ph } be a family of partitions of the domain Ω into straight-sided elements, and let V h ⊂ V be the associated standard finite element spaces of continuous piecewise polynomials of certain degree. Corresponding to the partition Ph , we use the symbols hK , hγ , E(K), Eh , Eh,0 as introduced in Section 3.2. In addition, we use Eh,ΓN , Eh,ΓC with obvious meanings. The discrete formulation of the variational inequality (3.98) reads: find uh ∈ V h such that a(uh , v h − uh ) + j(v h ) − j(uh ) ≥ (v h − uh )

∀ vh ∈ V h .

(3.101)

Like the continuous variational inequality (3.98), the discrete problem (3.101) has a unique solution uh ∈ V h and it is characterized by the existence of λhτ ∈ (L∞ (ΓC ))d such that Z a(uh , v h ) + g λhτ · v hτ ds = (v h ) ∀ vh ∈ V h , (3.102) ΓC

|λhτ | ≤ 1,

λhτ · uhτ = |uhτ |

a.e. on ΓC ,

(3.103)

where v hτ and uhτ denote the tangential components of v h and uh , respectively. Analysis of the finite element approximation of such problems in the general context of variational inequalities is extensively discussed in [34, 35]. In the context of finite element approximations of a problem more general than the one considered in this chapter, one can find in [46, Section 8.2] an optimal-order a priori error estimate under additional solution regularity, and a convergence result without any additional solution regularity assumption. Similar to the result in Section 3.4, we have the residual type error estimate X 2 2 2 keh k2V ≤ C ηR , ηR = ηR,K , K∈Ph

where 2 ηR,K = h2K krK k20;K +

+ hK

X

1 hK 2

γ∈E(K)∩Eh,ΓN

X

γ∈E(K)∩Eh,0

kRγ k20;γ

kRγ k20;γ + hK

X

γ∈E(K)∩Eh,ΓC

kRγ k20;γ ,

and rK and Rγ are the interior and side residuals, respectively, defined by (σ h = Cε(uh )):

68

V. Bostan, W. Han

rK = Div σ h + f 1 ⎧ ⎨ − [σ h ν]γ Rγ = f 2 − σ h ν ⎩ −gλhτ − σ hτ

Moreover, ⎛

2 ηR ≤ C ⎝keh k2V +

+

X

K∈Ph

X

γ∈Eh,ΓC

in K ∈ Ph ,

if γ ∈ Eh,0 , if γ ∈ Eh,ΓN , if γ ∈ Eh,ΓC .

hγ kλτ − λhτ k20;γ

h2K kf 1 − f 1,K k20;K +

X

γ∈Eh,ΓC

⎞

hγ kλhτ − λhτ ,γ k20;γ ⎠

with discontinuous piecewise polynomial approximations f 1,K , λhτ ,γ of residuals f 1 , λhτ , respectively. The results in Section 3.5 on recovery-based error estimate can be similarly extended to the finite element solution of the frictional contact problem (3.98). We now present numerical results on two two-dimensional problems. In both examples, body forces are assumed to be negligible and the body is in bilateral frictional contact with a rigid foundation on the part ΓC . The friction is modeled by Tresca’s law with a given slip bound g. We use the adaptive algorithm stated at the beginning of Section 3.6, with μ = 1. Example 3.2. The physical setting of this example shown in Figure 3.13. The domain Ω = (0, 4) × (0, 4) is the cross-section of a three-dimensional linearly elastic body and plane stress condition is assumed. On the part ΓD = {4}×(0, 4) the body is clamped. Oblique tractions act on the part {0}×(0, 4) and the part (0, 4) × {4} is traction free. Thus ΓN = ({0} × (0, 4)) ∪ ((0, 4) × {4}). The contact part of the boundary is ΓC = (0, 4) × {0}.

x

2

ΓN

Ω ΓC rigid obstacle

Fig. 3.13 Problem setting for Example 3.2.

ΓD

x1


69

Fig. 3.14 Initial mesh.

The elasticity tensor C satisfies (Cε)ij =

Eν E (ε11 + ε22 )δij + εij , 1 − ν2 1+ν

1 ≤ i, j ≤ 2,

where E is the Young’s modulus, ν is the Poisson’s ratio of the material, and δij is the Kronecker symbol. We use the following data (the unit daN/mm2 stands for “decanewtons per square millimeter”): E = 1500 daN/mm2 , ν = 0.4, f 1 = (0, 0) daN/mm2 , f 2 (x1 , x2 ) = (150(5 − x2 ), −75) daN/mm2 , g = 450 daN/mm2 .

The initial uniform triangulation P1 (128 elements, 81 nodes) is shown in Figure 3.14 with the interval [0, 4] being divided into 4/h equal parts, h = 1/2. Then the triangulation is successively halved, and the numerical solution corresponding to h = 1/64 is taken as the “true” solution u. To have an idea of the convergence behavior of the discrete Lagrange multipliers, we compute the errors kλτ − λhτ k0;ΓC corresponding to the sequence of uniform refinements. Here, λτ is the Lagrange multiplier corresponding to the parameter h = 1/64. Figure 3.15 provides a comparison of the errors 1/2 ku − uun kλτ − λhτ k0;ΓC . The numerical convergence order of h kV and h h1/2 kλτ − λhτ k0;ΓC is obviously higher than that of ku − uun h kV , indicating that the second term in the efficiency bound is expected to be of higher order compared to the first term ku−uh k2V . Note that in the two-dimensional case, λτ and λhτ are scalar-valued functions, also denoted by λ and λh . The (approximate) calculation of λh follows the procedure described in Section 3.6. Graphs of λh,1 and λh,2 with h = 1/64 are provided in Figures 3.16 and 3.17.

70

V. Bostan, W. Han

0

10

solution lambda

−1

10

1

Error

0.46

1

0.82

−2

10

−3

10

2

3

10


4

10

un 1/2 kλ Fig. 3.15 Errors ku − u un hτ − λ hτ k0,ΓC (4). h kV (¤ ) versus h

We use an adaptive procedure based on both residual type and recovery type estimates to obtain a sequence of approximate solutions uad h . The deformed configuration and the adaptive finite element mesh after four adaptive iterations are shown in Figures 3.18 (based on the residual type estimator, 5583 elements, 2921 nodes) and 3.19 (based on the recovery type estimator, 5437 elements, 2832 nodes). Figures 3.20 and 3.21 contain the error values ad ku − uun h kV and ku − uh kV . We observe a substantial improvement of the efficiency using adaptively refined meshes. Figures 3.22 and 3.23 provide the P 2 1/2 values of ηI = ( K ηI,K ) , I ∈ {R, G}, where ηI,K are computed using either residual type estimator (I = R) or recovery type estimator (I = G) on both uniform and adapted meshes. Table 3.3 contains the values of CI computed for uniform and adaptive solutions: CI =

ku − uh kV , ηI

I ∈ {R, G}.


1/2 2.63e-04 2.63e-04 8.73e-04 8.73e-04

1/4 2.39e-04 2.39e-04 8.05e-04 8.07e-04

1/8 2.18e-04 2.22e-04 7.46e-04 7.46e-04

1/16 1.93e-04 2.01e-04 6.90e-04 6.83e-04

(3.104)


71

1.2

1

0.8

λh,1

0.6

0.4

0.2

0

−0.2

0

0.5

1

1.5

0

0.5

1

1.5

2

2.5

3

3.5

4

2

2.5

3

3.5

4

x1

Fig. 3.16 Plot of λh,1 . 1.2

1

0.8

λh,2

0.6

0.4

0.2

0

−0.2

x1

Fig. 3.17 Plot of λh,2 .

Additional numerical experiments have been carried out in order to show the influence of the discrete Lagrange multipliers λh on the adaptive solution. To this end, we associate the residuals Rγ corresponding to the sides γ ∈ Eh,ΓC with a weighting parameter Θ: X 2 2 = ηR,K,Θ , ηR,Θ K∈Ph

2 ηR,K,Θ

= h2K krK k20;K +

1 hK 2

+ hK

X

γ∈E(K)∩Eh,0

X

γ∈E(K)∩Eh,ΓN

+ Θ hK

X

kRγ k20;γ kRγ k20;γ

γ∈E(K)∩Eh,ΓC

kRγ k20;γ .

72


4

3.5

3

2.5

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 3.18 Deformed configuration based on residual type estimator. 4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 3.19 Deformed configuration based on recovery type estimator.


73

0

10

Error


−1

10

1 0.46

0.63

1 −2

10

2

10

3

4


10

ad Fig. 3.20 Residual type estimator: ku − u un h kV (¤ ) versus ku − u h kV (4).

0

10

Error


−1

10

1 0.46

0.63

1 −2

10

2

10

3


4

10

ad Fig. 3.21 Recovery type estimator: ku − u un h kV (¤ ) versus ku − u h kV (4).

74

V. Bostan, W. Han

3

10

1 0.37 0.47

2

10

1

1

Error

10

0

10

−1

10

1 uniform mesh error uniform mesh ηR adapted mesh error adapted mesh ηR

−2

10

2

0.46 0.63 1 3

10

4


10


3

10

2

1

10

0.4 0.51 1 1

Error

10

0

10

−1

10

1 uniform mesh error uniform mesh ηG adapted mesh error adapted mesh ηG

−2

10

2

10

0.46 0.63 1 3



4

10


75

Table 3.4 Residual type estimator. Influence of Lagrange multiplier part on adaptive solution Θ=1 adaptive level 0 1 2 3 4 d.o.f. 162 368 930 2356 5842 ηR,Θ 6.49e+02 4.48e+02 2.96e+02 1.93e+02 1.26e+02 ku − u ad 1.71e-01 1.07e-01 6.57e-02 3.87e-01 2.19e-02 h kV Θ = 0.1 d.o.f. ηR,Θ ku − u ad h kV

162 366 924 2342 5836 6.41e+02 4.47e+02 2.96e+02 1.93e+02 1.25e+02 1.71e-01 1.08e-01 6.58e-02 3.86e-01 2.18e-02

Θ = 0.01 d.o.f. ηR,Θ ku − u ad h kV

162 366 920 2318 5716 6.40e+02 4.47e+02 2.96e+02 1.93e+02 1.26e+02 1.71e-01 1.08e-01 6.60e-02 3.86e-01 2.21e-02

We observe that the smaller the parameter Θ is, the less influence there is from the Lagrange multiplier part on the error estimator and hence on the adaptive mesh. An adaptive procedure is performed based on the error indicators ηR,Θ with Θ = 1, 0.1, 0.01. Numerical results are summarized in Table 3.4. Similar numerical results are obtained for the gradient recovery type estimator. t u Example 3.3. The physical setting of this example is shown in Figure 3.24. The domain Ω = (0, 10) × (0, 2) is the cross-section of a three-dimensional linearly elastic body with the plane stress condition assumed. On the part ΓD = (0, 10) × {2} the body is clamped. Horizontal tractions act on the part {0} × (0, 2) and oblique tractions on {10} × (0, 2). Here ΓC = (0, 10) × {0}.

Γ

x

D

2

x

Γ

N

Ω ΓC rigid obstacle

Fig. 3.24 Problem setting for Example 3.3.

ΓN

1

76

V. Bostan, W. Han


The following data are used: E = 1000 daN/mm2 , ν = 0.3, f 1 = (0, 0) daN/mm2 , ½ (500, 0) daN/mm2 on {0} × (0, 2), f 2 (x1 , x2 ) = (250x2 − 750, −100) daN/mm2 on {10} × (0, 2), g = 175 daN/mm2 .

We start with a coarse uniform triangulation shown in Figure 3.25, with 160 triangular elements and 105 nodes. For uniform triangulations, we divide the interval [0, 10] into 10/h equal parts and interval [0, 2] into 4/h parts. The numerical solution corresponding to h = 1/64 is taken as the “true” solution u. Figures 3.26 (based on the residual type estimator, 5222 elements, 2755 nodes) and 3.27 (based on the recovery type estimator, 4964 elements, 2624 nodes) show the approximate solution and refined mesh after four consecutive refinements. Graphs of λh,1 and λh,2 with h = 1/64 are provided in ad Figures 3.28 and 3.29. Again, we compute the errors ku−uun h kV , ku−uh kV , 1/2 h kλτ − λhτ k0;ΓC , and ηI , I ∈ {R, G}, whose values are provided in Figures 3.30—3.34. Table 3.5 contains the values of CI , I ∈ {R, G} defined in (3.104). t u Table 3.5 Numerical values of CR and CG h un CR ad CR un CG ad CG

1/2 4.60e-04 4.60e-04 1.39e-03 1.39e-03

1/4 4.27e-04 4.21e-04 1.29e-03 1.29e-03

1/8 3.86e-04 3.83e-04 1.22e-03 1.20e-03

1/16 3.24e-04 3.27e-04 1.15e-03 1.12e-03


77

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0

1

2

3

4

5

6

7

8

9

10

9

10

Fig. 3.26 Deformed configuration based on residual type estimator.

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0

1

2

3

4

5

6

7

8

Fig. 3.27 Deformed configuration based on recovery type estimator.

78


1

λh,1

0.5

0

−0.5

−1

−1.5

0

1

2

3

4

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

x1

Fig. 3.28 Plot of λh,1 . 1.5

1

λh,2

0.5

0

−0.5

−1

−1.5

x

1

Fig. 3.29 Plot of λh,2 .

0

10


1

Error

0.42 −1

10

0.66 1

−2

10

2

10

3


4

10

ad Fig. 3.30 Residual type estimator: ku − u un h kV (¤ ) versus ku − u h kV (4).


0

10


1

Error

0.42 −1

10

0.65 1

−2

10

2

10

3

4


10

ad Fig. 3.31 Recovery type estimator: ku − u un h kV (¤ ) versus ku − u h kV (4).

0

10

solution lambda

1 0.42

−1

Error

10

1

0.82 −2

10

−3

10

2

10

3


un 1/2 kλ Fig. 3.32 ku − u un hτ − λ hτ k0,ΓC (4). h kV (¤ ) versus h

4

10

79

80

V. Bostan, W. Han

3

10

1 0.29 0.37 2

1

10

1

Error

10

0

10

1 0.42

−1

10

uniform mesh error uniform mesh ηR adapted mesh error adapted mesh ηR

−2

10

2

0.66 1

3

10

4


10


3

10

1 2

0.37

10

0.51 1 1

Error

10

0

10

1 0.42

−1

10


−2

10

2

10

0.65 1

3



4

10


81

3.8 Quasistatic Variational Inequalities and Their Discretizations The discussions in the previous sections on steady-state problems can be extended in adaptive solution of time-dependent variational inequalities. We illustrate this on some quasistatic variational inequalities. We first introduce an abstract quasistatic variational inequality. As is seen below, this variational inequality covers several quasistatic contact problems. A similar quasistatic variational inequality also arises in the study of a primal formulation of some plasticity problems [42]. Let X be a real Hilbert space with inner product (·, ·)X and norm k · kX , T > 0, and 1 < p ≤ ∞. Assume a : X × X → R is a symmetric, continuous, coercive bilinear form, j : X → R is a continuous semi-norm X. Let there be given f ∈ W 1,p (0, T ; X) and u0 ∈ X with the condition a(u0 , v) + j(v) ≥ (f (0), v)X

∀ v ∈ X.

We consider the following variational inequality. Problem 3.1. Find u : [0, T ] → X such that for a.e. t ∈ (0, T ), a(u(t), v − u(t)) ˙ + j(v) − j(u(t)) ˙ ≥ (f (t), v − u(t)) ˙ X

∀ v ∈ X,

(3.105)

and u(0) = u0 .

(3.106)

A proof of the following well-posedness result can be found in [46]. Theorem 3.11. Under the stated assumptions on the data, Problem 3.1 has a unique solution u ∈ W 1,p (0, T ; X). Next, we consider a semidiscrete S scheme for Problem 3.1. We need a parN tition of the time interval: [0, T ] = n=1 [tn−1 , tn ] with 0 = t0 < t1 < · · · < tN = T . Denote kn = tn − tn−1 for the length of the subinterval [tn−1 , tn ], and k = maxn kn for the maximal step-size. For the given linear functional f ∈ W 1,p (0, T ; X) and the solution u ∈ W 1,p (0, T ; X), we use the notations fn = f (tn ) and un = u(tn ), which are well-defined by the Sobolev embedding W 1,p (0, T ; X) → C([0, T ]; X). The symbol δn un = (un − un−1 )/kn is used to denote the backward divided difference. Then a semidiscrete approximation of Problem 3.1 is the following. Problem 3.2. Find uk = {ukn }N n=0 ⊂ X such that uk0 = u0

(3.107)

and for n = 1, 2, . . . , N , a(ukn , v − δn ukn ) + j(v) − j(δn ukn ) ≥ (fn , v − δn ukn )X

∀ v ∈ X.

(3.108)

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We notice that under the assumptions of Theorem 3.11, Problem 3.2 has a unique solution. Let us briefly indicate how an error estimate can be derived for the semidiscrete solution defined in Problem 3.2. For this purpose, we assume u ∈ H 2 (0, T ; X). Note that this assumption in particular implies that the solution u satisfies the inequality (3.105) for all t ∈ (0, T ). Denote en = un − ukn for the solution error, and let kvka = a(v, v)1/2 be the “energy” norm over X. We observe that from the assumptions on the bilinear form a, the quantity kvka defines an equivalent norm on X. We take v = δn ukn in (3.105) at t = tn , v = u˙ n in (3.108), and add the two inequalities to obtain a(un , δn ukn − u˙ n ) + a(ukn , u˙ n − δn ukn ) ≥ 0. This relation can be rewritten as a(en , en − en−1 ) ≤ kn a(en , δn un − u˙ n ). Now, 1 2 1 a(en , δn un − u˙ n ) ≤ 2 a(en , en − en−1 ) ≥

Hence,

¢ ¡ ken k2a − ken−1 k2a ,

¢ ¡ ken k2a + kδn un − u˙ n k2a .

ken k2a ≤ (1 − kn )−1 ken−1 k2a + kn (1 − kn )−1 kδn un − u˙ n k2a . It is easy to verify that when k ≤ 1/2, (1 − kn )−1 ≤ e2 kn . Therefore, ken k2a ≤ e2 kn ken−1 k2a + kn e2 kn kδn un − u˙ n k2a . An inductive argument leads to ken k2a ≤ e2 (kn +···+k1 ) ke0 k2a +

n−1 X j=0

kn−j e2 (kn +···+kn−j ) kδn−j un−j − u˙ n−j k2a .

Because e0 = 0 and kn + · · · + kn−j ≤ T , we have ken k2a ≤ c

n−1 X j=0

kn−j kδn−j un−j − u˙ n−j k2a .

Under the smoothness assumption u ∈ H 2 (0, T ; X), we have


83

kδn−j un−j − u˙ n−j k2a ≤ kn−j k¨ uk2L2 (tn−j−1 ,tn−j ;X) . Thus, we have shown the error estimate N X max kun − ukn k2X ≤ c kj2 k¨ uk2L2 (tj−1 ,tj ;X) . 1≤n≤N

(3.109)

j=1

It then follows that

max kun − ukn kX ≤ c k k¨ ukL2 (0,T ;X) .

1≤n≤N

Note that the estimate (3.109) identifies the local contribution from each subinterval to the solution error, and therefore could be used to adjust the values of kj to achieve an efficient time discretization. Now we describe some contact problems that lead to a variational inequality of the type stated in Problem 3.1. We consider a quasistatic contact process of a linearly elastic body occupying a Lipschitz domain Ω in Rd (d ≤ 3 in applications). The boundary Γ = ∂Ω is partitioned as follows:Γ = ΓD ∪ ΓN ∪ ΓC with ΓD , ΓN , and ΓC relatively open and mutually disjoint, and meas(ΓD ) > 0. The time interval of interest is (0, T ). We assume that the body is clamped on ΓD × (0, T ), meaning that the body is clamped on ΓD during the time interval (0, T ); a volume force of density f 1 acts in Ω × (0, T ) and a surface traction of density f 2 is applied on ΓN × (0, T ). Then the displacement field u : Ω × [0, T ] → Rd and the stress field σ : Ω × [0, T ] → Sd satisfy the relations: σ = Cε(u) ε(u) = 12 (∇u + (∇u)T ) Div σ + f 1 = 0 u=0 σν = f 2 u(0) = u0

in Ω × (0, T ), in Ω × (0, T ), in Ω × (0, T ), on ΓD × (0, T ), on ΓN × (0, T ), in Ω.

(3.110) (3.111) (3.112) (3.113) (3.114) (3.115)

In (3.110), C : Ω × Sd → Sd denotes the fourth-order elasticity tensor of the material, and is assumed to be bounded, symmetric, and positive definite in Ω. In (3.115), u0 is the given initial displacement field. In (3.114), ν is the unit outward normal vector on the boundary Γ , which exists a.e. because Γ is assumed to be Lipschitz continuous. The relations (3.110)—(3.115) are to be supplemented by contact conditions on ΓC × (0, T ). We assume the contact is bilateral (no loss of contact during the process) and the friction is modeled with Tresca’s friction law (see, e.g., [29, 59]): ⎫ uν = 0, |σ τ | ≤ g, ⎬ |σ τ | < g ⇒ u˙ τ = 0, on ΓC × (0, T ). (3.116) ⎭ |σ τ | = g ⇒ ∃ λ ≥ 0 s.t. σ τ = −λu˙ τ

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V. Bostan, W. Han

Here, g ≥ 0 represents the friction bound function, uν and uτ are the normal and tangential components of u on Γ , and σ τ is the tangential component of σ. To introduce a variational formulation of the problem, we need the space V = {v ∈ (H 1 (Ω))d : v = 0 a.e. on ΓD , vν = 0 a.e. on ΓC } with the inner product and norm defined by Z (u, v)V = Cε(u) : ε(v) dx, Ω

kvkV =

p (v, v)V .

Assume that the volume force and traction densities satisfy f 1 ∈ W 1,∞ (0, T ; (L2 (Ω))d ),

f 2 ∈ W 1,∞ (0, T ; (L2 (ΓN ))d ),

and let the constant g > 0 be given. We define over the space V , Z a(u, v) = Cε(u) : ε(v) dx, Ω Z g |v τ | ds j(v) =

(3.117)

(3.118) (3.119)

ΓC

and denote by l(t) the element of V given by Z Z (l(t), v)V = f 1 (t) · v dx + f 2 (t) · v ds Ω

ΓN

∀ v ∈ V, t ∈ [0, T ]. (3.120)

We assume that the initial data satisfy u0 ∈ V, a(u0 , v) + j(v) ≥ (l(0), v)V

(3.121) (3.122)

∀ v ∈ V.

Then a standard procedure leads to the following variational formulation for the frictional contact problem (3.110)—(3.115) and (3.116). Problem 3.3. Find u : [0, T ] → V such that for a.e. t ∈ (0, T ), ˙ ˙ ˙ a(u(t), v − u(t)) + j(v) − j(u(t)) ≥ (l(t), v − u(t)) V

∀ v ∈ V, (3.123)

and u(0) = u0 .

(3.124)

We observe that this is a special case of Problem 3.1. By Theorem 3.11, we see that under assumptions (3.117), (3.121), and (3.122), Problem 3.3 has a unique solution u ∈ W 1,∞ (0, T ; V ). Some other contact conditions lead to a similar variational inequality. For example, suppose the contact condition is described by a simplified version


85

of Coulomb’s law (see, e.g., [29, 59]): ⎫ σν = S, |σ τ | ≤ μ |σν |, ⎬ |σ τ | < μ |σν | ⇒ u˙ τ = 0, ⎭ |σ τ | = μ |σν | ⇒ ∃ λ ≥ 0 s.t. σ τ = −λu˙ τ

on ΓC × (0, T ).

(3.125)

Here S ∈ L∞ (ΓC ) is a given function on ΓC and μ ∈ L∞ (ΓC ) is the given coefficient of friction, μ ≥ 0 a.e. on ΓC . Then the variational formulation is still of the form (3.123)—(3.124) with the same bilinear form a and V = {v ∈ (H 1 (Ω))d : v = 0 a.e. on ΓD }, Z μ |S| |v τ | ds, j(v) = ΓC Z Z Z f 1 (t) · v dx + f 2 (t) · v ds + (l(t), v)V = Ω

ΓN

S vν ds.

ΓC

Analysis of this variational inequality is similar to that for the problem (3.123)—(3.124). We now show how to derive a posteriori error estimates for finite element solutions of the temporally semidiscrete problem. We choose Problem 3.3 as the model quasistatic contact problem. Its temporal semidiscrete approximation is the following. k k Problem 3.4. Find uk = {ukn }N n=0 , where un ∈ V , 0 ≤ n ≤ N , u0 = u0 , such that for n = 1, 2, . . . , N ,

a(ukn , v − δn ukn ) + j(v) − j(δn ukn ) ≥ (ln , v − δn ukn )

∀ v ∈ V,

(3.126)

where ln = l(tn ). Denote wkn = δn ukn . Then ukn = ukn−1 + kn wkn and we can express the inequality problem (3.126) in terms of wkn : Find wkn ∈ V such that kn a(wkn , v − wkn ) + j(v) − j(wkn )

≥ (ln , v − wkn ) − a(ukn−1 , v − wkn )

∀ v ∈ V. (3.127)

This inequality is equivalent to the minimization problem: wkn ∈ V,

Jn (wkn ) = inf Jn (v),

(3.128)

v ∈V

where Jn is the functional Jn (v) =

kn a(v, v) + j(v) − (ln , v) + a(ukn−1 , v), 2

v ∈ V.

(3.129)

We now turn to a finite element approximation of Problem 3.4. We use the finite element space setting discussed in Section 3.7. For a given v ∈ (L1 (Ω))d ,

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V. Bostan, W. Han

similar to (3.10)—(3.11), we define the interpolation operator Πh : V → V h as follows: R X e v ψa dx Πh v = v a ϕa , where v a = RKa . (3.130) e a ϕa dx K a ∈N v,0

Applying Theorem 3.2, we have an h-independent constant C > 0 such that for all v ∈ V and f ∈ (L2 (Ω))d , Z

Ω

X

γ∈Eh

f · (v − Πh v) dx ≤ C|v|1;Ω ⎝

X

a ∈Nh,0

kh−1 K (v

h2a min kf − f a k20;Ke ⎠ , a f ∈Rd a

(3.132)

(3.133)

kh−1/2 (v − Πh v)k20;γ ≤ C|v|21;Ω . γ

(3.134)

−

Πh v)k20;K

(3.131) ⎞1/2

C|v|21;Ω ,

K∈Ph

X

|v − Πh v|21;Ω ≤ C|v|21;Ω , ⎛

≤

The finite element approximation of the inequality problem (3.126) is to h such that find uhk n ∈V h hk h hk h hk a(uhk n , v − δn un ) + j(v ) − j(δn un ) ≥ (ln , v − δn un )

∀ vh ∈ V h. (3.135) Corresponding to the formulation (3.127), the finite element method is to h find whk such that n ∈V h hk h hk kn a(whk n , v − w n ) + j(v ) − j(wn )

hk h hk ≥ (ln , v h − whk n ) − a(un−1 , v − w n )

∀ v h ∈ V h . (3.136)

As in previous sections, wkn is characterized by the existence of a unique λknτ ∈ L∞ (ΓC ) such that Z k kn a(wn , v) + gλknτ · v τ ds = (ln , v) − a(ukn−1 , v) ∀ v ∈ V, (3.137) ΓC

|λknτ |

≤ 1 a.e. on ΓC

and

λknτ · wknτ = |wknτ | a.e. on ΓC ;

(3.138)

hk ∞ whk n is characterized by the existence of a unique λnτ ∈ L (ΓC ) such that h kn a(whk n ,v ) +

Z

ΓC

h gλhk nτ · v τ ds

|λhk nτ | ≤ 1 a.e. on ΓC

h = (ln , v h ) − a(uhk n−1 , v )

and

∀ v h ∈ V h , (3.139)

hk hk λhk nτ · w nτ = |w nτ | a.e. on ΓC .

(3.140)


87

3.9 A Posteriori Error Estimates for the Quasistatic Contact Problem We first provide a dual formulation for the problem (3.127). Let Q = (L2 (Ω))d×d × (L2 (ΓC ))d . For n = 1, 2, . . . , N , define Fn : V × Q → R by the formula ¸ Z ∙ kn k Cq 1 : q 1 − f 1n · v + Cε(un−1 ) : ε(v) dx Fn (v, q) = 2 Ω Z Z + g |q 2 | ds − f 2n · v ds, ΓC

ΓN

where q = (q 1 , q 2 ) ∈ Q, f 1n = f 1 (tn ), and f 2n = f 2 (tn ). Introduce a linear bounded operator Λ : V → Q by the relation Λv = (ε(v), v τ ) Then for any v ∈ V ,

∀ v ∈ V.

Jn (v) = Fn (v, Λv),

and the minimization problem (3.6) can be rewritten as wkn ∈ V,

Fn (wkn , Λwkn ) = inf Fn (v, Λv). v ∈V

(3.141)

Let V ∗ and Q∗ = (L2 (Ω))d×d × (L2 (ΓC ))d be the duals of V and Q placed in duality by the pairings h·, ·iV and h·, ·iQ , respectively. We need to compute the conjugate function Fn∗ of Fn : Fn∗ (Λ∗ q ∗ , −q ∗ ) ≡

sup v ∈V,q∈Q

{hΛ∗ q ∗ , viV − hq ∗ , qiQ − Fn (v, q)} ,

where Λ∗ : Q∗ → V ∗ is the adjoint of Λ. We have ½Z ∙ ∗ ∗ ∗ ∗ Fn (Λ q , −q ) = sup (q ∗1 − Cε(ukn−1 )) : ε(v) + f 1n · v v ∈V,q∈Q

+

Z

Ω

¶¸ kn Cq 1 : q 1 + q ∗1 : q 1 dx 2 ¾ Z f 2n · v ds + [q ∗2 · v τ − (q ∗2 · q 2 + g |q 2 |)] ds . −

ΓN

µ

ΓC

(3.142)

Let A : Ω × Sd → Sd be the fourth-order tensor, inverse to C. Like C, A is also bounded, symmetric, and positive definite in Ω. Then we can show that

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V. Bostan, W. Han

Fn∗ (Λ∗ q ∗ , −q ∗ ) =

⎧Z ⎨

1 Aq ∗1 : q 1 dx 2k n Ω ⎩ +∞

if q ∗ ∈ Q∗f,g , otherwise,

where the admissible function set Q∗f,g consists of all q ∗ = (q 1 , q 2 ) ∈ Q∗ such that |q ∗2 | ≤ g a.e. on ΓC and Z Z Z ¤ £ ∗ f 2n · v ds + q ∗2 · v τ ds (q 1 − Cε(ukn−1 )) : ε(v) + f 1n · v dx + Ω

ΓN

=0

ΓC

∀ v ∈ V.

(3.143)

The dual problem of (3.141) is to find p∗ ∈ Q∗f,g such that −Fn∗ (Λ∗ p∗ , −p∗ ) =

sup q ∗ ∈Q∗ f,g

©

ª − Fn∗ (Λ∗ q ∗ , −q ∗ ) .

(3.144)

Applying Theorem 3.1, we know that both problems (3.141) and (3.144) have unique solutions and the following duality relation holds: Fn (wkn , Λwkn ) = −Fn∗ (Λ∗ p∗ , −p∗ ).

(3.145)

k As in Section 3.3, we first let whk n ∈ V be any approximation of w n . By using (3.127) and (3.129) we obtain

kn 2 hk k kwkn − whk n kV = Jn (w n ) − Jn (wn ). 2 Let p∗ be the solution of the dual problem (3.144). Relation (3.145) implies Jn (wkn ) = F (wkn , Λwkn ) = −Fn∗ (Λ∗ p∗ , −p∗ ) ≥ −Fn∗ (Λ∗ q ∗ , −q ∗ )

∀ q ∗ ∈ Q∗f,g .

Therefore, for any q ∗ = (q ∗1 , q ∗2 ) ∈ Q∗f,g and r∗ ∈ (L2 (Ω))d×d , kn 2 kwkn − whk n kV 2

Z Z 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ≤ Jn (whk ) + F (Λ q , −q ) + Ar : r dx − Ar∗ : r∗ dx n n 2k 2k n n Ω Ω Z 1 hk ∗ hk ∗ C(kn ε(wn ) + Ar ) : (kn ε(wn ) + Ar ) dx = Ω 2kn Z Z £ ¤ k hk hk Cε(un−1 ) : ε(wn ) − f 1n · wn dx + f 2n · whk + n ds Ω ΓN Z Z Z 1 ∗ ε(whk g |whk (Aq ∗1 : q ∗1 − Ar∗ : r ∗ ) dx. − n ) : r dx + nτ | ds + 2k n Ω ΓC Ω (3.146)


89

It follows immediately from (3.143) that for any q ∗ = (q ∗1 , q ∗2 ) ∈ Q∗f,g , Z

q ∗1

: ε(v) dx +

Ω

Z

ΓC

q ∗2

· v τ ds =

Z

¤ £ Cε(ukn−1 ) : ε(v) − f 1n · v dx Ω Z f 2n · v ds ∀ v ∈ V. (3.147) − ΓN

Using (3.147) and regrouping terms in (3.146) we find that for any q ∗ ∈ Q∗f,g and r∗ ∈ (L2 (Ω))d×d , Z kn 1 k hk 2 ∗ hk ∗ C(kn ε(whk kwn − wn kV ≤ n ) + Ar ) : (kn ε(w n ) + Ar ) dx 2 Ω kn Z 1 A(q ∗1 − r∗ ) : (q ∗1 − r∗ ) dx + k Ω n Z ¡ ¢ ∗ hk + g |whk nτ | + q 2 · w nτ ds. ΓC

Thus, for any r∗ ∈ (L2 (Ω))d×d , Z kn 1 2 ∗ hk ∗ k ≤ Ckn (ε(whk kwkn − whk n V n ) + Ar ) : (kn ε(w n ) + Ar ) dx 2 k n Ω ½Z 1 + ∗ inf∗ A(q ∗1 − r∗ ) : (q ∗1 − r∗ ) dx q ∈Qf,g Ω kn ¾ Z ¡ ¢ hk ∗ hk g |wnτ | + q 2 · wnτ ds . (3.148) + ΓC

The second term ½Z ¾ Z ¡ ¢ 1 ∗ ∗ ∗ ∗ hk ∗ hk II ≡ ∗ inf∗ A(q 1 − r ) : (q 1 − r ) dx + g |wnτ | + q 2 · wnτ ds q ∈Qf,g Ω kn ΓC

on the right-hand side of estimate (3.148) is bounded as follows. First, from the definition (3.143), the term II equals ½Z Z ¡ ¢ 1 ∗ ∗ ∗ ∗ ∗ hk sup A(q − r ) : (q − r ) dx + inf g |whk 1 1 nτ | + q 2 · w nτ ds ∗ ∗ q ∈Q v ∈V Ω kn ΓC |q ∗ 2 |≤g Z ¤ £ ∗ (q 1 − Cε(ukn−1 )) : ε(v) + f 1n · v dx + ¾ ZΩ Z ∗ + f 2n · v ds + q 2 · v τ ds . ΓN

ΓC

Here and below, the condition “|q ∗2 | ≤ g” stands for “|q ∗2 | ≤ g a.e. on ΓC .” Substitute q ∗1 − r∗1 by q ∗1 and regroup the terms to see that II equals

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¸ ½Z ∙ kn ∗ k inf sup − Cε(v) : ε(v) + (r − Cε(un−1 )) : ε(v) + f 1n · v dx 4 |q ∗ 2 |≤g v ∈V Ω ¾ Z Z Z ¡ ¢ ∗ hk ∗ hk + f 2n · v ds + q 2 · v τ ds + g |wnτ | + q 2 · wnτ ds . ΓN

ΓC

ΓC

Define the residual

Rn (q ∗2 , r∗ ) = sup v ∈V

1 kvkV

½Z

¤ £ ∗ (r − Cε(uhk n−1 )) : ε(v) + f 1n · v dx Ω ¾ Z Z + f 2n · v ds + q ∗2 · v τ ds . (3.149) ΓN

ΓC

Then ½ kn II ≤ inf sup − kvk2V + (Rn (q ∗2 , r ∗ ) + kukn−1 − uhk n−1 kV )kvkV |q ∗ |≤g 4 v ∈V 2 ¾ Z ¡ ¢ hk ∗ hk + g |wnτ | + q 2 · wnτ ds ΓC ½ 1 2 = inf (Rn (q ∗2 , r∗ ) + kukn−1 − uhk n−1 kV ) k |q ∗ |≤g n 2 ¾ Z ¡ ¢ hk ∗ hk + g |wnτ | + q 2 · wnτ ds . ΓC

Therefore, we have the following result.

Theorem 3.12. Let wkn ∈ V be the unique solution of (3.127), and whk n ∈V an approximation. Then for any r∗ ∈ (L2 (Ω))d×d , the following error bound Z kn 1 2 ∗ hk ∗ k ≤ C(kn ε(whk kwkn − whk n V n ) + Ar ) : (kn ε(w n ) + Ar ) dx 2 Ω kn ½ 1 2 + inf (Rn (q ∗2 , r∗ ) + kukn−1 − uhk n−1 kV ) ∗ kn |q 2 |≤g ¾ Z ¡ ¢ ∗ hk g |whk ds + | + q · w nτ 2 nτ ΓC

is valid, where the residual Rn (q ∗2 , r∗ ) is defined by (3.31).

Theorem 3.12 provides a general framework for various a posteriori error estimates with different choices of the auxiliary variable r∗ . From now ∗ on, whk n is the finite element solution defined by (3.136). We choose r = hk ∗ ∗ −kn Cε(wn ), and then replace the infimum over q 2 by taking q 2 = −gλhk nτ . Then Theorem 3.12 leads to the following corollary. Corollary 3.1. Let ukn ∈ V be the unique solution of (3.126) and uhk n ∈ V h its finite element approximation defined by (3.135). Then there exists a constant C such that the following a posteriori error estimate holds:


91

k hk kukn − uhk n kV ≤ CRn + Ckun−1 − un−1 kV ,

(3.150)

where Rn ≡ sup v ∈V

1 kvkV

½Z

Ω

+

Z

¤ £ Cε(uhk n ) : ε(v) − f 1n · v dx − gλhk nτ

ΓC

¾ · v τ ds .

Z

ΓN

f 2n · v ds (3.151)

Let Πh be the interpolation operator defined by (3.130). Substituting v by v − Πh v in (3.151) we obtain ½Z ¤ £ 1 Rn ≡ sup Cε(uhk n ) : ε(v − Πh v) − f 1n · (v − Πh v) dx kvk V v ∈V Ω ¾ Z Z hk f 2n · (v − Πh v) ds + gλnτ · (v − Πh v)τ ds . − ΓN

ΓC

Decompose the integrals into local contributions from each element K ∈ Ph and apply Green’s formula over K to find ⎧ Z 1 ⎨ X Rn = sup (−Div σ(uhk n ) − f 1n ) · (v − Πh v) dx v ∈V kvkV ⎩ K K∈Ph

+

X Z

γ∈Eh,0

+

X

γ∈Eh,ΓN

+

γ

[σ(uhk n )ν] · (v − Πh v) ds

Z

γ

¡ ¢ σ(uhk n )ν − f 2n · (v − Πh v) ds

⎫ ⎬ ´ X Z ³ hk ) + gλ v) ds , σ(uhk · (v − Π h τ nτ n τ ⎭ γ

γ∈Eh,ΓC

(3.152)

where we used decomposition σν · v = σν vν + σ τ · v τ = σ τ · v τ a.e. on ΓC . Define the interior residuals for each element K ∈ Ph by rK = −Div σ(uhk n ) − f 1n and side residuals for each side γ ∈ Eh by ⎧ [σ(uhk n )ν] ⎪ ⎨ hk Rγ = σ(un )ν − f 2n ⎪ ⎩ hk σ(uhk n )τ + gλnτ

in K,

(3.153)

if γ ∈ Eh,0 ,

if γ ∈ Eh,ΓN , if γ ∈ Eh,ΓC .

(3.154)

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V. Bostan, W. Han

Note that residuals corresponding to the sides lying on ΓD are considered to be 0. By using definitions (3.153) and (3.154), relation (3.152) reduces to ⎧ ⎫ Z ⎬ XZ 1 ⎨ X Rn = sup rK · (v − Πh v) dx + Rγ · (v − Πh v) ds . ⎭ v ∈V kvkV ⎩ K γ K∈Ph

γ∈Eh

(3.155) Using the estimates (3.133) and (3.134) in (3.155), and applying the Cauchy— Schwarz inequality, we get ⎧ ⎫ 1⎬ ⎨ ³ ´ X X 1 2 Rn ≤ sup C kvkV h2K krK k20;K + hγ kRγ k20;γ ⎭ v ∈V kvkV ⎩ K∈Ph

⎛

≤C⎝

X

K∈Ph

h2K krK k20;K +

γ∈Eh

X

γ∈Eh

⎞ 12

hγ kRγ k20;γ ⎠ .

(3.156)

We summarize the above results in the form of a theorem. h Theorem 3.13. Let ukn ∈ V and uhk n ∈ V be the unique solutions of (3.126) and (3.135), respectively. Then the following a posteriori error estimate holds:

⎛

⎝ kukn − uhk n kV ≤ C

X

K∈Ph

h2K krK k20;K +

X

γ∈Eh

+ C kukn−1 − uhk n−1 kV ,

⎞ 12

hγ kRγ k20;γ ⎠

(3.157)

where rK and Rγ are interior and, respectively, side residuals, defined by (3.153) and (3.154). In practical computations, the terms on the right side of (3.157) are regrouped by writing kukn − uhk n kV ≤ C

Ã

X

2 ηK,R

K∈Ph

! 12

+ Ckukn−1 − uhk n−1 kV ,

(3.158)

where the local error indicator ηK,R on each element K, defined by 2 ηK,R = h2K krK k20;K +

1 2

X

γ∈E(K)∩Eh,0

hγ kRγ k20;γ +

X

γ∈E(K)∩Eh,Γ

hγ kRγ k20;γ ,

(3.159) identifies contributions from each of the elements to the global error. Similar to the derivation given in the second half of Section 3.4, we have the following inequality on the efficiency of the error estimate:


X

2 ηK,R ≤C

K∈Ph

Ã

2 kukn − uhk n kV +

+

X

h2K kf 1n

K∈Ph

+

X

X

2 hγ kλknτ − λhk nτ k0;γ

γ∈Eh,ΓC

− f 1n,K k20,K +

hk 2 hγ kλhk nτ − λnτ ,γ k0,γ

γ∈Eh,ΓC

93

!

X

hγ kf 2n − f 2n,γ k20,γ

γ∈Eh,ΓN

(3.160)

with discontinuous piecewise polynomial approximations f 1n,K , f 2n,γ , and hk λhk nτ ,γ of f 1n , f 2n , and λnτ . We can apply Theorem 3.12 to derive a stress recovery type a posteriori error estimate, and we restrict our discussion to linear elements. Similar to (3.58)—(3.59), we define the stress recovery operator Gh : V h → (V h )d as follows: Z X 1 h ∗ h ∗ ∗ Gh σ(v ) = σ (v ) = σ a ϕa , σa = σ(v h ) dx. (3.161) e a | Ke a | K a ∈Nh

In the case of linear elements

σ ∗a =

Na X

i αa (σ(v h ))Kai ,

(3.162)

i=1

where (σ(v h ))Kai denotes the tensor value of the stress σ(v h ) on the element e a = SNa Kai , αia = |Kai |/|K e a |, i = 1, . . . , Na . Kai , K i=1 hk hk Let un and un−1 be the finite element solutions defined by (3.135) at hk the time steps n and n − 1, respectively. Let r∗ = −σ ∗ (uhk n ) + σ(un−1 ) and hk ∗ ∗ ∗ hk replace the infimum over q 2 by taking q 2 = −gλnτ , where σ (un ) is the recovered stress defined by (3.161) and λhk nτ is provided by Theorem 3.17. With these choices, estimate (3.32) becomes Z kn2 2 ∗ hk hk ∗ hk k ≤ (σ(uhk kwkn − whk n V n ) − σ (un )) : (ε(un ) − ε (un )) dx 2 Ω ¢2 ¡ + Rn + kukn−1 − uhk n−1 kV ¡ ¢2 ∗ hk 2 k hk ≤ Ckσ(uhk , n ) − σ (un )k0;Ω + Rn + kun−1 − un−1 kV (3.163) ∗ hk where ε∗ (uhk n ) = Aσ (un ) and the residual Rn is given by ½Z Z £ ∗ hk ¤ 1 Rn ≡ sup σ (un ) : ε(v) − f 1n · v dx − f 2n · v ds v ∈V kvkV Ω ΓN ¾ Z hk + g λnτ · v τ ds . (3.164) ΓC

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V. Bostan, W. Han

We can then obtain the following result. h Theorem 3.14. Let ukn ∈ V and uhk n ∈ V be the unique solutions of (3.126) and (3.135), respectively. Then the following a posteriori error estimate holds:

kukn − uhk n kV ≤ C

Ã

X

2 ηK,G

K∈Ph

! 12

⎛

+C⎝

X

a ∈Nh,0

+ Ckukn−1 − uhk n−1 kV ,

⎞ 12

h2a min kf 1n − f a k20;Ke ⎠ f a ∈Rd

a

(3.165)

where local indicators ηK,G are computed for every element K ∈ Ph by X 2 ∗ hk 2 2 ηK,G = kσ(uhk hγ kσ ∗ (uhk n ) − σ (un )k0;K + n )ν − f 2n k0;γ γ∈E(K)∩Eh,ΓN

X

+

γ∈E(K)∩Eh,ΓC

hk 2 hγ kσ ∗ (uhk n )τ + gλnτ k0;γ .

We observe that if f 1n ∈ (L2 (Ω))d then Ã

X

2

a ∈Nh

ha min kf 1n − f a ∈Rd

f a k20;Ke a

and if f 1n ∈ (H 1 (Ω))d , then Ã

X

2

a ∈Nh

ha min kf 1n − f a ∈Rd

f a k20;Ke a

(3.166)

!1/2

!1/2

= o(h),

= O(h2 ).

P 2 )1/2 is a reliTheorem 3.14 asserts that the estimator ηG = ( K∈Ph ηK,G k hk able upper bound of a constant multiple of the error kun − un kV . We can also show the following inequality: ⎛ X X 2 2 2 ηK,G ≤ C ⎝kukn − uhk hγ kλknτ − λhk n kV + nτ k0;γ K∈Ph

γ∈Eh,ΓC

+

X

K∈Ph

+

h2K kf 1n − f 1n,K k20,K +

X

γ∈Eh,ΓC

⎞

X

γ∈Eh,ΓN

hk 2 ⎠ , hγ kλhk nτ − λnτ ,γ k0,γ

hγ kf 2n − f 2n,γ k20,γ

(3.167)

with discontinuous piecewise polynomial approximations f 1n,K , f 2n,γ , and hk λhk nτ ,γ of f 1n , f 2n , and λnτ .


95

3.10 Numerical Example on the Quasistatic Contact Problem We follow the procedure described in Section 3.6 for the finite element approximation. In order to show the effectiveness of the adaptive procedure we compare numerical convergence orders of the approximate solutions. We compute these orders by considering families of uniform and locally refined triangulations. N Consider a sequence of finite element solutions {uhk n,un }n=0 based on uniform triangulations of the domain Ω and the same uniform partition of the time interval. Starting with an initial coarse triangulation P1 , we construct a family of nested meshes by the subdivision of each triangle into four triangles. The solution from the most refined mesh is taken as the “true” solution {ukn }N n=0 that is used to compute the errors of the approximate solutions obtained on the other meshes. N The finite element solution {uhk n,ad }n=0 is obtained using the following adaptive algorithm: 1. Start with the initial triangulation Ph and corresponding finite element subspace V h . N hk h 2. Compute the finite element solution {uhk n,ad }n=1 , where un,ad ∈ V , 0 ≤ n ≤ N. 3. At the time step tN and for each element K ∈ Ph compute the error estimator ηK,I P of residual type (I = R) or gradient recovery type (I = G). 4. Let η = ( K∈Ph ηK,I )/Ne , where Ne is the total number of elements. An element K is marked for refinement if ηK > μ η, where μ is a prescribed threshold. In our example, μ = 1. 5. Perform refinement and obtain a new triangulation Ph . 6. Return to step 2. Example 3.4. We consider the physical setting shown in Figure 3.35. The domain Ω = (0, 2)×(0, 10) is the cross-section of a three-dimensional linearly

Γ

x

D

2

x

Ω

ΓN

ΓC rigid obstacle

Fig. 3.35 Setting of the problem.

Γ

N

1

96

V. Bostan, W. Han


elastic body and plane stress condition is assumed. On the part ΓD = (0, 10)× {2} the body is clamped. Oblique tractions act on the parts {0} × (0, 2) and {10} × (0, 2). Thus ΓN = ({0} × (0, 2)) ∪ ({10} × (0, 2)). The contact part of the boundary is ΓC = (0, 10) × {0}. The elasticity tensor C satisfies (Cε)ij =

Eν ( 1 − ν2

11

+

22 )δij

+

E 1+ν

ij ,

1 ≤ i, j ≤ 2,

where E is the Young’s modulus, ν is the Poisson’s ratio of the material, and δij is the Kronecker symbol. We use the following data: E = 1000 daN/mm2 ,

ν = 0.3,

2

f 1 = (0, 0) daN/mm , f 2 (x1 , x2 , t) = (2.5t, 0) daN/mm2

if (x1 , x2 ) ∈ {0} × (0, 2),

f 2 (x1 , x2 , t) = (2(x2 − 2)t, −t) daN/mm2 2

g = 1 daN/mm ,

u0 = 0m,

T = 1 sec.

if (x1 , x2 ) ∈ {10} × (0, 2),

The time step considered is t = 0.2 sec. We start with the initial triangulation P1 (160 elements, 105 nodes) shown in Figure 3.36. Here, the interval [0, 1] is divided into 1/h equal parts with h = 1/2 which is successively halved. The numerical solution corresponding to h = 1/64 is taken as the “true” solution {ukn }N n=0 . To have an idea of the convergence behaviour of the discrete Lagrange multipliers, we compute the errors maxn kλknτ − λhk nτ k0;ΓC corresponding to the sequence of uniform refinements. Here, {λknτ }N n=0 is the Lagrange multiplier corresponding to the parameter h = 1/64. Figure 3.43 1/2 provides the relative error values maxn kukn − uhk maxn kλknτ − n,un kV and h 1/2 maxn kλknτ − λhk λhk nτ k0;ΓC . The numerical convergence order of h nτ k0;ΓC is k hk obviously higher than that of maxn kun −un,un kV , indicating that the second term in the efficiency bounds (3.160) and (3.167) is expected to be of higher


97

2 order compared to the first term kukn − uhk n kV . Graphs of λh,1 and λh,2 with h = 1/64 are provided in Figures 3.39 and 3.40. We use an adaptive procedure based on both residual type and gradient recovery type estimates to obtain a sequence of approximate solutions N {uhk n,ad }n=0 . The deformed configuration and the adaptive finite element mesh after four adaptive iterations are shown in Figure 3.37 (based on the residual type estimator, 4792 elements, 2531 nodes) and Figure 3.38 (based on the recovery type estimator, 4894 elements, 2583 nodes). Figures 3.41 and 3.42 conk hk tain the relative error values maxn kukn − uhk n,un kV and maxn kun − un,ad kV . We observe a substantial improvement of the efficiency using adaptively reP 2 1/2 fined meshes. Figures 3.44 and 3.45 provide the values of ηI = ( K ηK,I ) , I ∈ {R, G}, where ηK,I are computed using either residual type estimator (I = R) or recovery type estimator (I = G) on both uniform and adapted meshes. Table 3.6 contains the values of CI computed for uniform and adaptive solutions:

CI = max n

kukn − uhk n kV , k kun kV ηI

I ∈ {R, G}.

t u


1/2 9.29e-02 9.29e-02 3.18e-01 3.18e-01

1/4 8.48e-02 8.37e-02 2.99e-01 2.94e-01

1/8 8.27e-02 7.99e-02 2.99e-01 2.92e-01

1/16 8.12e-02 7.56e-02 2.99e-01 2.77e-01

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0

1

2

3

4

5

6

7

8

9

10

Fig. 3.37 Residual type estimator. Deformed configuration (amplified by 100) at t = 1 sec.

98

V. Bostan, W. Han 2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0

1

2

3

4

5

6

7

8

9

10

Fig. 3.38 Recovery type estimator. Deformed configuration (amplified by 100) at t = 1 sec. 1.5

1

λ

h,1

0.5

0

−0.5

−1

−1.5

0

1

2

3

4

5

x

6

7

8

9

10

6

7

8

9

10

1

Fig. 3.39 Plot of λh 1 at t = 1 sec. 1.5

1

λ

h,2

0.5

0

−0.5

−1

−1.5

0

1

2

Fig. 3.40 Plot of λh 2 at t = 1 sec.

3

4

5

x1


99

0

10


Error

1

0.4

−1

10

0.59

1 2

10

3

4


10

k hk Fig. 3.41 Residual type estimator. maxn ku kn −u hk n,un kV (¤ ) versus maxn ku n −u n,ad kV (4). 0

10


Error

1

0.4

−1

10

0.61

1 2

10

3


4

10

k hk Fig. 3.42 Recovery type estimator. maxn ku kn −u hk n,un kV (¤ ) versus maxn ku n −u n,ad kV (4).

100

V. Bostan, W. Han 0

10

solution lambda

1 0.4 −1

Error

10

1 −2

10

0.58

2

3

10

4


10

1/2 max kλ k − λ hk k Fig. 3.43 maxn ku kn − u hk n n,un kV (¤ ) versus h nτ nτ 0,ΓC (4). 1

10

1 0.38

0

10 Error

0.53 1

−1

10

uniform mesh error uniform mesh ηR adapted mesh error adapted mesh ηR 2

10

3


4

10

Fig. 3.44 Residual type estimator. maxn ku kn − u hk n,un kV and ηR on uniform mesh (¤ ) k and η on adapted mesh (4). versus maxn ku kn − u hk V R n,ad


101


0

10

1 Error

0.4

0.54 1

−1

10

2

10

3

4


10

Fig. 3.45 Recovery type estimator. maxn ku kn − u hk n,un kV and ηG on uniform mesh (¤ ) versus maxn ku kn − u hk n,ad kV and ηG on adapted mesh (4). 0

10

Error

adapted mesh (residual) adapted mesh (gradient recovery)

1 0.61

−1

10

0.59

1 2

10

3


Fig. 3.46 Residual type estimator versus recovery type estimator.

4

10

102

V. Bostan, W. Han

3.11 Concluding Remarks This chapter presents a general framework on a posteriori error estimation for finite element solutions of some variational inequalities, through the employment of the duality theory in convex analysis. The general error estimate contains an auxiliary variable. Different choices of the auxiliary variable leads to different a posteriori error bounds; in particular, residual type and recovery type error estimates are shown and analyzed in this chapter. The a posteriori error estimates are used in developing adaptive finite element algorithms for the variational inequalities. The effectiveness of the adaptive finite element algorithms is demonstrated in several numerical examples. Due to the inequality nature of the problems, efficiency bounds for the error estimators contain terms related to approximations of Lagrange multipliers; see, for example, (3.57) and (3.82). Currently, sharp bounds for such terms are unknown. Ideally, it is hoped that terms such as X hγ kλ − λh k20;γ γ∈Eh,Γ2

can be bounded by o(h2 ), and then efficiency of the error estimators can be rigorously deduced from (3.57) and (3.82). Numerical results reported in the chapter strongly suggest the terms involving the Lagrange multiplier approximations are of higher order, and performance of the adaptive finite element algorithms is not sensitive with respect to the calculation of the Lagrange multipliers. The derivation and analysis of the a posteriori error estimates are done on some model problems in this chapter, and can be extended to other or more general variational inequalities, including those arising in frictional contact mechanics. Another direction for future research is to develop adaptive algorithms for simultaneous time and space discretizations of time-dependent variational inequalities, and their applications in solving evolutionary problems in contact mechanics. Acknowledgments We thank Professor David Y. Gao for various helpful suggestions on this chapter. This work was supported by NSF under grant DMS-0106781.

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75. O.C. Zienkiewicz, Origins, milestones and directions of the finite element method–A personal view, in P.G. Ciarlet and J.-L. Lions, eds., Handbook of Numerical Analysis, Vol. IV, North-Holland, Amsterdam, 1996, pp. 3—67. 76. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol. 1 (Basic Formulation and Linear Problems), McGraw-Hill, New York, 1989. 77. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol. 2 (Solid and Fluid Mechanics, Dynamics and Nonlinearity), McGraw-Hill, New York, 1991. 78. O.C. Zienkiewicz and J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Meth. Eng. 24 (1987), 337—357. 79. O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Meth. Eng. 33 (1992), 1331—1364. 80. O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. Numer. Meth. Eng. 33 (1992), 1365—1382.

Chapter 4

Time—Frequency Analysis of Brain Neurodynamics W. Art Chaovalitwongse, W. Suharitdamrong, and P.M. Pardalos

Summary. The characteristics of neurodynamics of intracranial electroencephalogram (EEG) at different frequency bands were investigated in a sample of two patients with epilepsy. The results indicate a tendency for the gamma, theta, and alpha frequency bands in EEG signals to have a higher dimensional complexity than the beta and gamma frequency bands. We also investigate the time—frequency component decomposition of EEG signals and observe very different perceptual complexity and a difference in evoked spectral responses, which could be a reflection of neuronal recruitment that triggers the epileptogenic process. The results of this study may provide insights to the brain network’s mechanism by which local and regional circuits can continuously form and reform with different regions functionally disconnected from other brain areas. Key words: EEG, brain dynamics, time—frequency distribution, chaos theory, optimization

4.1 Introduction The electroencephalogram (EEG) measures brainwaves of different frequencies within specific areas of the brain. Electrodes are implanted and placed W. Art Chaovalitwongse Department of Industrial and Systems Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ, e-mail: [email protected] W. Suharitdamrong Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, e-mail: [email protected] P.M. Pardalos Departments of Industrial and Systems Engineering, Computer Science, and Biomedical Engineering, University of Florida, Gainesville, FL, e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization, Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_4, © Springer Science+Business Media, LLC 2009

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on specific sites in the temporal lobe or the scalp to monitor and record the electrical impulses and neuronal activities in the brain. The extent of the noise effect depends on whether the additive noise caused by the remote electrodes is sufficient to induce important variations of the dynamics across the electrodes. It is well known that electrocortical activity exhibits microchaos while macroscopically behaving as a linear, near-equilibrium system. For this reason, EEG time series recorded at macroscopic scale may exhibit chaos, and it is common for physical phenomena to exhibit different dynamic features at different scales. Characterizing EEG signals by using measures of chaos derived from the theory of nonlinear dynamics has been proven to accurately reflect underlying attractor properties. The study of nonlinear dynamics involves the statistical investigation of molecular motion, sound waves, the weather, and other dynamical systems. As EEG exhibits particular dynamic properties at particular scales, which can be called far-from-equilibrium cellular and small cellular group interactions, the properties of EEG contrast with near-equilibrium dynamics and linear wave superposition at the macroscopic level. In this research, we focus on a question of whether the macroscopic EEG time series preserves chaotic dynamical features carried through from the underlying neural events driving the wave motion and how to apply a statistical analysis to study neurodynamics of EEG time series. Before one could claim to understand the neurodynamics of cortical neurons, one needs to understand how their dynamics differ with scale and how they interact across scale associated with the brain electrical activity. In this research, we direct our application to the epilepsy research. We investigate the neurodynamics of the EEG and its sensitivity to different frequency bands linked to epileptic waveforms in the brain of patients with epilepsy.

4.1.1 The Fact About Epilepsy and EEG Epilepsy is the second most common serious brain disorder after stroke. Worldwide, at least 40 million people or 1% of the population currently suffer from epilepsy. Epilepsy is a chronic condition of diverse etiologies with the common symptom of spontaneous recurrent seizures, which is characterized by intermittent paroxysmal and highly organized rhythmic neuronal discharges in the cerebral cortex. Seizures can temporarily disrupt normal brain functions such as motor control, responsiveness, and recall, which typically last from seconds to a few minutes. There is a localized structural change in neuronal circuitry within the cerebrum which produces organized quasirhythmic discharges in some types of epilepsy (i.e., focal or partial epilepsy). These discharges then spread from the region of origin (epileptogenic zone) to activate other areas of the cerebral hemisphere. Nonetheless, the mechanism by which these fixed disturbances in local circuitry produce intermittent disturbances of brain function is not well comprehended. The development of the

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epileptic state can be considered as changes in network circuitry of neurons in the brain. When neuronal networks are activated, they produce changes in voltage potential, which can be captured by EEG. These changes are reflected by wriggling lines along the time axis in a typical EEG recording. A typical electrode montage for EEG recordings in our study is shown in Figure 4.1.

Fig. 4.1 Inferior transverse views of the brain, illustrating approximate depth and subdural electrode placement for EEG recordings. Subdural electrode strips are placed over the left orbitofrontal (LOF), right orbitofrontal (ROF), left subtemporal (LST), and right subtemporal (RST) cortex. Depth electrodes are placed in the left temporal depth (LTD) and right temporal depth (RTD) to record hippocampal activity.

The model for macroscopic neurodynamics depends on the epileptiforms in EEG time series (with the presence of noise) at different frequency bands. The effect of electrical activities at different frequency bands upon microscopic neural activity of high chaoticity level of the signal can be used to portray the mutual information between macroscopic pools of neurons in the brain. As instantaneous natural frequencies, damping factors, and coupling coefficients describing the dynamics of small pools of coupled neurons are stochastically independent, macroscopic neuronal wave motions are assumed to obey superposition and position near-equilibrium. This assumption leads to the intriguing possibility that local coherence among pools of neuronal cells might be generated from subsets of cells currently engaged in high-amplitude microscopic chaos, the local pool then providing a driving signal to the linear and near-equilibrium macroscopic dynamics. This mechanism appears to offer a further means of bridging dynamic interactions across scale. For this reason, the research is motivated to apply the nonlinear measures based on the theory of nonlinear dynamics as they have been proved capable of capturing the microscopic and macroscopic chaos in EEG time series.

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Fig. 4.2 Twenty second EEG recording of the pre-ictal state of a typical epileptic seizure obtained from 32 electrodes. Each horizontal trace represents the voltage recorded from electrode sites listed in the left column (see Figure 4.1 for anatomical location of electrodes).

Fig. 4.3 Twenty second EEG recording of the post-ictal state of a typical epileptic seizure obtained from 32 electrodes. Each horizontal trace represents the voltage recorded from electrode sites listed in the left column (see Figure 4.1 for anatomical location of electrodes).

4.1.2 EEG Frequency Bands The EEG measures brainwaves of different frequencies within the brain. Electrodes are placed on specific sites on the scalp to detect and record the electrical impulses within the brain as shown in Figures 4.1 to 4.3.


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The raw EEG has usually been described in terms of frequency bands: Delta (0.1—3 Hz), Theta (4—8 Hz), Alpha (8—12 Hz), Beta (13—30 Hz), and Gamma (above 30 Hz).

Delta (0.1—3 Hz) The EEG Delta waves are less than 4 Hz and occur in deep sleep and in some abnormal processes also during experiences of “empathy state.” Delta waves are involved with our ability to integrate and release. As it reflects the unconscious mind, the information in our unconscious mind can be accessed through Delta. It is the dominant rhythm in infants up to one year of age and it is present in stages 3 and 4 of sleep. The EEG Delta waves tend to be the highest in amplitude and the slowest waves. However, most individuals diagnosed with attention deficit disorder, naturally increase rather than decrease Delta activity when trying to focus. The inappropriate Delta response often severely restricts the ability to focus and maintain attention. It is as if the brain is locked into a perpetual drowsy state [8].

Theta (4—8 Hz) Theta activity has a frequency of 4 to 8 Hz and is classified as “slow” activity. It can be seen in connection with creativity, intuition, daydreaming, and fantasizing and is a repository for memories, emotions, and sensations. Theta waves are strong during internal focus, meditation, prayer, and spiritual awareness. It reflects the state between wakefulness and sleep. Theta is believed to reflect activity from the limbic system and hippocampal regions. Theta is observed in anxiety, behavioral activation, and behavioral inhibition [8].

Alpha (8—12 Hz) Alpha waves are those between 8 and 12 Hz. Good healthy alpha production promotes mental resourcefulness, aids in the ability to mentally coordinate, and enhances an overall sense of relaxation and fatigue. When Alpha predominates most people feel at ease and calm. Alpha also appears to bridge the conscious to the subconscious. Alpha rhythms are reported to be derived from the white matter of the brain. The white matter can be considered the part of the brain that connects all parts with each other. Alpha is a common state for the brain and occurs whenever a person is alert, but not actively processing information. It can also be used as a marker for alertness and sleep. Alpha has been linked to extroversion (introverts show less), creativity (creative subjects show alpha when listening and coming to a solution for cre-

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ative problems), and mental work. Alpha is one of the brain’s most important frequencies for learning and using information taught in the classroom and on the job [8].

Beta (13—30 Hz) Beta activity is considered to be “fast” activity. It reflects desynchronized active brain tissue. It is usually seen on both sides in symmetrical distribution and is most evident frontally; however, it may be absent or distorted in areas of cortical damage. Beta activity is generally regarded as a normal rhythm and is the dominant rhythm in those who are alert or anxious or who have their eyes open. It is the state that most of brain is in when we have our eyes open and are listening and thinking during analytical problem solving, judgment, decision making, or processing information about the world around us [8].

Gamma (above 30 Hz) Gamma activity is the only frequency group found in every part of the brain. When the brain needs to simultaneously process information from different areas, it is hypothesized that the 40 Hz activity consolidates the required areas for simultaneous processing. A good memory is associated with wellregulated and efficient 40 Hz activity, whereas a 40 Hz deficiency creates learning disabilities [8].

4.1.3 Chaos in Brain The term “chaos” in the theory of nonlinear dynamics is associated with exponential divergence of trajectories in phase space, which can reflect sensitivity to initial conditions. The evolution of chaos theory over the past two decades has mostly dealt with apparently simple systems with few degrees of freedom that can exhibit chaotic behavior. The brain is certainly a complex high-dimensional system, which has been proven to exhibit high-dimensional chaos associated with many brain state variables. The theory of chaos in brain appears that phenomena characteristic of many complex nonlinear systems (e.g., self-organization, interactions at multiple temporal and spatial scales, stable spatial structure in the presence of temporal chaos, etc.) occur in neocortex and may be closely aligned with cognitive processing. One of the most important concepts in the brain theories is that linear or quasilinear phenomena at one scale can coexist with highly nonlinear phenomena at another


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scale. This belief lies in communicating several general concepts, which are apparently critical to brain dynamic function, to disparate fields. A proof of concept about the nonstationarity of EEG time series comes from that fact that although locally chaotic EEG dynamics with globally linear dynamics does not address evidence for nonstationary temporal signals in limit set dimension, in this study we deal with multichannel EEG time series and embed the time series in higher dimension, which makes globally chaotic EEG to be nonlinear dynamics in spatiotemporal aspects. This concept suggests that the dynamical properties of EEG time series in our model demonstrate significant large-scale temporal and spatial heterogeneity in cortical function, which can produce macroscopic chaos. Our approaches based on the theory of nonlinear dynamics seem to address the evidence for nonstationarity of EEG time series as a function of spatiotemporal correlation of coupled oscillators. A macroscopic model of EEG should address some issues that arise specifically at the transition from locally chaotic dynamics to the macroscopic scale. In this study, we consider the system of nonlinear EEG dynamics, including standing waves with frequency in the 10 Hz range (within a factor of about two or three), which will give us insights on the increasing frequency with maturation of the alpha rhythm and negative correlation between brain size and alpha frequency during seizure episodes. The organization of the succeeding sections of this chapter is as follows. The background of previous studies in neocortical dynamics, neurodynamics of the brain, and the measures of chaoticity are described in Section 4.2. In Section 4.3, the design of experiment in this study is described as well as the computational methods to test the hypothesis. The results are described in Section 4.4. The conclusions and discussion are addressed in Section 4.5.

4.2 Neurodynamics of the Brain In the last decade, time series analysis based on chaos theory and the theory of nonlinear dynamics, which are among the most interesting and growing research topics, has been applied to time series data with some degree of success. The concepts of chaos theory and the theory of nonlinear dynamics have not only been useful to analyze specific systems of ordinary differential equations or iterated maps, but have also offered new techniques for time series analysis. Moreover, a variety of experiments has shown that a recorded time series is driven by a deterministic dynamical system with a low-dimensional chaotic attractor, which is defined as the phase space point or set of points representing the various possible steady-state conditions of a system, an equilibrium state or group of states to which a dynamical system converges. Thus, the theories of chaos and nonlinear dynamics have provided new theoretical and conceptual tools that allow us to capture, understand, and link the complex behaviors of simple systems. Characterization and quantification of the

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dynamics of nonlinear time series are also important steps toward understanding the nature of random behavior and may enable us to predict the occurrences of some specific events that follow temporal dynamical patterns in the time series. Several quantitative system approaches incorporating statistical technique nonlinear methods based on chaos theory have been successfully used to study epilepsy because the aperiodic and unstable behavior of the epileptic brain is suitable to nonlinear techniques that allow precise tracking of the temporal evolution. Our previous studies have shown that seizures are deterministic rather than random. Consequently, studies of the spatiotemporal dynamics in long-term intracranial EEGs, from patients with temporal lobe epilepsy, demonstrated the predictability of epileptic seizures; that is, seizures develop minutes to hours before clinical onset. The period of a seizure’s development is called a pre-ictal transition period, which is characterized by gradual dynamical changes in EEG signals of critical electrode sites approximately 1/2 to 1 hour duration before the ictal onset [4, 22, 17, 20, 14, 18, 15, 26, 29, 34, 33]. During a pre-ictal transition period, gradual dynamical changes can be exposed by a progressive convergence (entrainment) of dynamical measures (e.g., short-term maximum Lyapunov exponents, ST Lmax ) at specific anatomical areas and cortical sites, in the neocortex and hippocampus. Another measure we have used in the state space created from the EEG at individual electrode sites in the brain, average ¯ has produced promising results too. The value of Ω ¯ angular frequency (Ω), quantifies the average rate of the temporal change in the state of a system and is measured in rads/sec. Although the existence of the pre-ictal transition period has recently been confirmed and further defined by other investigators [9, 10, 23, 31, 24, 29], the characterization of this spatiotemporal transition is still far from complete. For instance, even in the same patient, a different set of cortical sites may exhibit a pre-ictal transition from one seizure to the next. In addition, this convergence of the normal sites with the epileptogenic focus (critical cortical sites) is reset after each seizure [4, 21, 18, 29]. Therefore, complete or partial post-ictal resetting of pre-ictal transition of the epileptic brain, affects the route to the subsequent seizure, contributing to the apparently nonstationary nature of the entrainment process. In those studies, however, the critical site selections are not trivial but extremely important because most groups of brain sites are irrelevant to the occurrences of the seizures and only certain groups of sites have dynamical convergence in the pre-ictal transition. Because the brain is a nonstationary system, algorithms used to estimate measures of the brain dynamics should be capable of automatically identifying and appropriately weighing existing transients in the data. In a chaotic system, orbits originating from similar initial conditions (nearby points in the state space) diverge exponentially (expansion process). The rate of divergence is an important aspect of the system dynamics and is reflected in the value of Lyapunov exponents and dynamical phase. Estimates of Lyapunov


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exponents and phase velocity of EEG time series are shown to be consistent with a global theory of EEG in which waves are partly transmitted along corticocortical fibers in the brain. General features of neocortical dynamics and their implications for theoretical descriptions are considered. Such features include multiple scales of interaction, multiple connection lengths, local and global time constants, dominance of collective interactions at most scales, and periodic boundary conditions. Epileptiforms and other epileptic activities in EEG time series may be directly related to the continuous forming and reforming of local and regional circuits that are functionally disconnected from tissue involved in global operation.

4.2.1 Estimation of Lyapunov Exponents The method we developed for estimation of short term largest Lyapunov exponents (ST Lmax ), an estimate of Lmax for nonstationary data, is explained in detail elsewhere [13, 16, 36]. Herein we present only a short description of our method. Construction of the embedding phase space from a data segment x(t) of duration T is made with the method of delays. The vectors Xi in the phase space (see Figure 4.4) are constructed as Xi = (x(ti ), x(ti + τ ) . . . x(ti + (p − 1) ∗ τ )),

(4.1)

Fig. 4.4 Diagram illustrating the estimation of ST Lmax measures in the state space. The fiducial trajectory, the first three local Lyapunov exponents (L1 , L2 , L3 ), is shown.

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where τ is the selected time lag between the components of each vector in the phase space, p is the selected dimension of the embedding phase space, and ti ∈ [1, T − (p − 1)τ ]. If we denote by L the estimate of the short-term largest Lyapunov exponent ST Lmax then: N

L= with

a 1 X |δXi,j (∆t)| log2 Na ∆t i=1 |δXi,j (0)|

δXi,j (0) = X(ti ) − X(tj ), δXi,j (∆t) = X(ti + ∆t) − X(tj + ∆t),

(4.2)

(4.3) (4.4)

where • X(ti ) is the point of the fiducial trajectory φt (X(t0 )) with t = ti , X(t0 ) = (x(t0 ), . . . , x(t0 +(p−1)∗τ )), and X(tj ) is a properly chosen vector adjacent to X(ti ) in the phase space (see below). • δXi,j (0) = X(ti ) − X(tj ) is the displacement vector at ti , that is, a perturbation of the fiducial orbit at ti , and δXi,j (∆t) = X(ti + ∆t) − X(tj + ∆t) is the evolution of this perturbation after time ∆t. • ti = t0 + (i − 1) ∗ ∆t and tj = t0 + (j − 1) ∗ ∆t, where i ∈ [1, Na ] and j ∈ [1, N ] with j 6= i. • ∆t is the evolution time for δXi,j , that is, the time one allows δXi,j to evolve in the phase space. If the evolution time ∆t is given in sec, then L is in bits per second. • t0 is the initial time point of the fiducial trajectory and coincides with the time point of the first data in the data segment of analysis. In the estimation of L, for a complete scan of the attractor, t0 should move within [0, ∆t]. • Na is the number of local Lmax s estimated within a duration T data segment. Therefore, if Dt is the sampling period of the time domain data, T = (N − 1)Dt = Na ∆t + (p − 1)τ . We computed the ST Lmax profiles using the method proposed by Iasemidis and coworkers [13], which is a modification of the method by Wolf et al. [36]. We call the measure short term to distinguish it from those used to study autonomous dynamical systems studies. Modification of Wolf’s algorithm is necessary to better estimate ST Lmax in small data segments that include transients, such as inter-ictal spikes. The modification is primarily in the search procedure for a replacement vector at each point of a fiducial trajectory. For example, in our analysis of the EEG, we found that the crucial parameter of the Lmax estimation procedure, in order to distinguish between the pre-ictal, the ictal, and the post-ictal stages, was not the evolution time ∆t nor the angular separation Vi,j between the evolved displacement vector δXi−1,j (∆t) and the candidate displacement vector δXi,j (0) (as was claimed in Frank et al. [11]). The crucial parameter is the adaptive estimation in time and


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9 8

Lmax (bits/sec)

7 6 5 4 3 2 1 0

25

50

75 100 Time (Minutes)

125

150

Fig. 4.5 Smoothed ST Lmax profiles over 2 hours derived from an EEG signal recorded at RTD2 (patient 1). A seizure (SZ 10) started and ended between the two vertical dashed lines. The estimation of the Lmax values was made by dividing the signal into nonoverlapping segments of 10.24 sec each, using p = 7 and τ = 20 msec for the phase space reconstruction. The smoothing was performed by a 10 point (1.6 min) moving average window over the generated ST Lmax profiles.

phase space of the magnitude bounds of the candidate displacement vector to avoid catastrophic replacements. Results from simulation data of known attractors have shown the improvement in the estimates of L achieved by using the proposed modifications [13]. In the pre-ictal state, depicted in Figure 4.5, one can see a trend of ST Lmax toward lower values over the whole preictal period, with one prominent drop in the value of ST Lmax approximately 24 minutes prior to the seizure (denoted by an asterisk in the figure). This pre-ictal drop in ST Lmax can be explained as an attempt of the system toward a new state of less degrees of freedom long before the actual seizure [17].

4.2.2 Estimation of EEG Phase Velocity Motivated by the representation of a state as a vector in the state space, another chaoticity measure employed in this research is a use of frequency-wave

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spectra to obtain phase velocity estimates. To estimate the phase velocity, we define the difference in phase between two evolved states X(ti ) and X(ti +∆t) as ∆Φi [19]. Then, denoting with (∆Φ) the average of the local phase differences ∆Φi between the vectors in the state space, we have: ∆Φ =

Nα 1 X · ∆Φi , Nα i=1

(4.5)

where Nα is the total number of phase differences estimated from the evolution of X(ti ) to X(ti + ∆t) in the state space, according to: ¯ ¯ ¯ X(ti ) · X(ti + ∆t) ¯¯ ¯ . (4.6) ∆Φi = ¯arccos kX(ti )k · kX(ti + ∆t)k ¯ ¯ is: Then, the average angular frequency Ω

¯ = 1 · ∆Φ. Ω ∆t

(4.7)

¯ is given in rad/sec. Thus, whereas ST Lmax If ∆t is given in sec, then Ω ¯ measures measures the local stability of the state of the system on average, Ω ¯ by how fast a local state of the system changes on average (e.g., dividing Ω −1 2π, the rate of change of the state of the system is expressed in sec = Hz). ¯ profile over time is given in Figure 4.6. The An example of a typical Ω values are estimated from a 60 minute long EEG sample recorded from an electrode located in the epileptogenic hippocampus. The EEG sample includes a 2 minute seizure that occurs in the middle of the recording. The state space was reconstructed from sequential, nonoverlapping EEG data segments of 2048 points (sampling frequency 200 Hz, hence each segment of 10.24 sec in duration) with p = 7 and τ = 4, as for the estimation of ST Lmax profiles [19]. The pre-ictal, ictal, and post-ictal states correspond ¯ respectively. The highest Ω ¯ values to medium, high, and lower values of Ω, ¯ were observed during the ictal period, and higher Ω values were observed during the pre-ictal period than during the post-ictal period. This pattern roughly corresponds to the typical observation of higher frequencies in the original EEG signal ictally, and lower EEG frequencies post-ictally. However, these observations can hardly denote a long-term warning of an impending seizure. The estimates of phase velocity of EEG data exhibit evidence for an underlying characteristic velocity of the EEG time series along the cortical surface. These results support the general theoretical idea that an EEG is composed of traveling waves, which are partly combined to form standing waves and propagated along corticocortical fibers. However, the results demonstrate that the existence of waves at such large scales does not preclude the simultaneous existence of waves at several smaller scales in which propagation can be adequately described in terms of exclusively intracortical interactions.


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Angular Frequency − Phase (rads/sec)

1.8

1.6

1.4

1.2

1

0.8

0.6 0

25

50

75 100 Time (minutes)

125

150

¯ profile before, during and after an epileptic seizure, estimated from Fig. 4.6 A typical Ω the EEG recorded from a site in the epileptogenic hippocampus; the seizure occurred between the vertical lines.

4.2.3 Spatiotemporal Dynamics ¯ profiles at individual cortical sites, Based on the estimated ST Lmax and Ω the temporal evolution of the stability of each cortical site is quantified. However, the system under consideration (brain) has a spatial extent and, as such, information about the transition of the system towards the ictal state should also be included in the interactions of its spatial components. The spatial dynamics of this transition are captured by consideration of the ¯ between different cortical sites. For exrelations of the ST Lmax (and Ω) ample, if a similar transition occurs at different cortical sites, the ST Lmax of the involved sites are expected to converge to similar values prior to the transition. We have called such participating sites “critical electrode sites.” We have used periods of 10 minutes (i.e., moving windows including approximately 60 ST Lmax values over time at each electrode site) to test the convergence at the 0.01 statistical significance level. We employed the T index (from the well-known paired T -statistics for comparisons of means) as a measure of distance between the mean values of pairs of ST Lmax profiles over time. The T -index at time t between electrode sites i and j is de-

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fined as Ti,j (t) =

√ N × |E{ST Lmax,i − ST Lmax,j }| /σi,j (t),

(4.8)

where E{·} is the sample average difference for the ST Lmax,i − ST Lmax,j estimated over a moving window wt (λ) defined as ½ 1 if λ ∈ [t − N − 1, t], wt (λ) = 0 if λ 6∈ [t − N − 1, t], where N is the length of the moving window. Then, σi,j (t) is the sample standard deviation of the ST Lmax differences between electrode sites i and j within the moving window wt (λ). The thus defined T -index follows a tdistribution with N − 1 degrees of freedom. For the estimation of the Ti,j (t) indices in our data we used N = 60 (i.e., average of 60 differences of ST Lmax exponents between sites i and j per moving window of approximately 10 minute duration). Therefore, a two-sided t-test with N − 1 (= 59) degrees of freedom, at a statistical significance level α should be used to test the null hypothesis, Ho : “brain sites i and j acquire identical ST Lmax values at time t.” In this experiment, we set α = 0.01, the probability of a type I error, or better, the probability of falsely rejecting Ho if Ho is true, is 1%. For the T -index to pass this test, the Ti,j (t) value should be within the interval [0,2.662].

4.2.4 Optimization in the Brain Neurodynamics Having quantified the spatiotemporal dynamics of the brain, we propose optimization techniques to identify the critical electrode sites. This problem can be naturally modeled as a quadratic 0—1 program, which has been extensively used to study Ising spin glass models [1, 2, 3, 12, 25]. Specifically, the critical electrode selection problem is formulated as a quadratic 0—1 knapsack problem with the objective function to minimize the average T -index (a measure of statistical distance between the mean values of ST Lmax ) among electrode sites and the knapsack constraint to identify the number of critical cortical sites. The problem is formally formulated as follows. Let A be an n × n matrix, whose each element ai,j represents the T index between electrode i and j within a 10 minute window before the onset of a seizure. Define x = (x1 , . . . , xn ), where each xi represents the cortical electrode site i. If the cortical site i is selected to be one of the critical electrode sites, then xi = 1; otherwise, xi = 0. A quadratic function is defined on Rn by min f (x) = xT Ax,

s.t. xi ∈ {0, 1}, i = 1, . . . , n,

(4.9)


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where A is an n × n matrix [27, 28]. Next, we add a linear constraint, Pn x i=1 i = k, where k is the number of critical electrode sites that we want to select. We now consider the following linearly constrained quadratic 0—1 problem: P¯ : min f (x) = xT Ax,

s.t.

n X

for some k, x ∈ {0, 1}n , A ∈ Rn×n .

xi = k

i=1

Problem P¯ can be formulated as a quadratic 0—1 problem of the form as in (4.9) by using an exact penalty. If A = (aij ) then let ⎤ ⎡ n n X X |aij |⎦ + 1. M = 2⎣ j=1 i=1

Then, we have the following equivalent problem P . P : min g(x) = xT Ax + M

Ã n X i=1

xi − k

!2

,

s.t. x ∈ {0, 1}n , A ∈ Rn×n .

Such a problem can be solved by applying a branch and bound algorithm with a dynamic rule for fixing variables [27, 28]. Previous studies by our group have shown the existence of resetting of the brain after seizure onset [35, 21, 32], that is, divergence of ST Lmax profiles after seizures. Therefore, to ensure that the optimal group of critical sites shows this divergence, we reformulate this optimization problem by adding one more quadratic constraint. The quadratically constrained quadratic zero— one problem is given by: min xT Ax Pn s.t. i=1 xi = k

xT Bx ≥ Tα k(k − 1),

(4.10) (4.11) (4.12)

where xi ∈ {0, 1} ∀i ∈ {1, . . . , n}. Note that the matrix B = (bij ) is the T -index matrix of brain sites i and j within 10 minute windows after the onset of a seizure. Tα is the critical value of the T -index, as previously defined, to reject Ho : “two brain sites acquire ¯ values within time window wt (λ).” identical Ω With one more quadratic constraint, the quadratic 0—1 problem becomes much harder to solve. Note that in the approach, a branch and bound algorithm with a dynamic rule for fixing variables cannot be applied to solve this problem because of the additional quadratic constraint [27, 28]. A conventional linearization has been proposed to solve this problem by introducing a new variable for each product of two variables and adding some

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additional constraints, and then formulating this problem as a mixed-integer linear (MILP) problem. Specifically, for each product xi xj , we introduce a new 0—1 variable, xij = xi xj (i 6= j). Note that xii = x2i = xi for xi ∈ {0, 1}. The equivalent MILP problem is given by XX min aij xij (4.13) i

s.t.

j

n X

xi = k,

(4.14)

i=1

xij ≤ xi ,

for i, j = 1, . . . , n (i 6= j)

(4.15)

xij ≤ xj ,

for i, j = 1, . . . , n (i 6= j)

(4.16)

xi + xj − 1 ≤ xij , for i, j = 1, . . . , n (i 6= j) XX bij xij ≥ Tα k(k − 1),

(4.17)

i

(4.18)

j

where xi ∈ {0, 1} and 0 ≤ xij ≤ 1, i, j = 1, . . . , n. Although the above-mentioned linearization technique can be used to solve the quadratically constrained quadratic integer program, a better linearization technique has been proposed [5]. The reason is that the above formulation is computationally inefficient as n increases. A more efficient MILP for electrode selection problem proposed in [5, 26] is given by min

n X si

(4.19)

i=1

s.t.

n X i=1

−

n X

xi − k = 0

(4.20)

aij xj + si + yi = 0,

for i = 1, . . . , n

(4.21)

yi − M (1 − xi ) ≤ 0,

for i = 1, . . . , n

(4.22)

0

hi − M xi ≤ 0,

for i = 1, . . . , n

(4.23)

dij xj + hi ≤ 0,

for i = 1, . . . , n

(4.24)

j=1

−

n X j=1

n X i=1

hi ≥ Tα k(k − 1),

(4.25)

where xi ∈ {0, 1} and si , yi , hi ≥ 0, for i, j = 1, . . . , n, and M 0 = kAk∞ and M = kBk∞ .


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4.3 Design of Experiments In this study, we investigate the properties of the brain neurodynamics from the continuous long-term multichannel intracranial EEG recordings that had been acquired from two patients with medically intractable temporal lobe epilepsy. The recordings were obtained as part of a pre-surgical clinical evaluation. They had been obtained using Nicolet BMSI 4000 and 5000 recording systems, using a 0.1 Hz high-pass and a 70 Hz low-pass filter. Each record included a total of 28 to 32 intracranial electrodes (8 subdural and 6 hippocampal depth electrodes for each cerebral hemisphere). In this framework, we filter the EEG data and then estimate ST Lmax and phase velocity, which measure the order or disorder of EEG signals recorded from individual electrode sites. Because the EEG consists of several frequency components (as described in the introduction) embedded in a single time series, bandpass filters are used to extract a particular frequency band from the EEG. In this experiment Butterworth filters are used to filter specified frequency bands (Delta, Theta, Alpha, Beta, and Gamma). In this study, we use the 10th order of Butterworth filters for each frequency band. Filter parameters were generated by using the butter command in MATLAB. Then the EEG from each channel was passed to each filter as shown in Figure 4.7. In these experiments, we aim to investigate if the mechanism of epileptogenesis implies a specific function for each frequency band response. Stimuli by epileptogenesis processes are likely to elicit dynamical changes in intracortical interactions. Such stimulation might trigger neurodynamical processes and we investigate whether additional processes will follow.

EEG Analysis Delta(0.5-4 Hz)

EEG Analysis Theta (4-8 Hz)

EEG Analysis EEG

Alpha (8-14 Hz)

EEG Analysis Beta (14-30 Hz)

Butterworth Bandpass Filter

Fig. 4.7 Filter design of the proposed analysis for each frequency band.

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4.3.1 Hypothesis I To determine whether synchronized and/or repetitive activity of neurons serves a specific epileptogenesis process, one must compare brain responses in two states (normal and abnormal) only one of which induces changes in neurodynamics. In this study, we speculate that the chaoticity level of different frequency bands of EEG data moving in the same direction lead to synchronous and fast oscillatory activity of neurons activated by the epileptogenesis stimuli. We hypothesize that filtering EEG data by using a different cutoff frequency might give different effects/synchronous and fast oscillatory activity of the EEG, which can be reflected from the average value of ST Lmax profiles. To test this hypothesis, we have to implement the narrowband lowpass filter (LPF) using an IIR filter. Note that designing this filter is a very difficult task (but very straightforward to implement). One of the reasons is that it is extremely hard for a filter to get such a sharp response. Another reason is that a perfectly linear phase (constant group delay) cannot be realized using IIR filtering. There are several approaches to designing an approximately linear phase IIR filter. A more widely used approach is to iteratively design filters that simultaneously minimize errors in magnitude and group delay. Another widely used approach is to design an IIR filter that approximates the desired magnitude response (e.g., an elliptic filter) and then design an IIR all-pass filter which compensates for the nonlinear phase. In this study, we implement IIR filtering procedures using the FD toolbox in MATLAB, which makes it possible to design a very good LPF magnitude response using an elliptic filter. This technique allows us to keep the poles within a circle of a specified radius. This approach employs a least-pth algorithm that attempts to minimize an Lp -norm error. The signal’s magnitude response of the Lp -norm error is given by: Z π 1 1 p |H|p = |H(ω) − Hd (ω)| W (ω) dω p , (4.26) 2π −π where H is the actual frequency response, Hd is the desired response, and W is some weighting function. In practice, the weighting function is normally equal to 1 over the passband and stopband and 0 in the transition band. Note that minimizing the L2 -norm is equivalent to minimizing the root mean square (RMS) error in the magnitude. In contrast, the L∞ -norm is equivalent to minimizing the maximum error over the frequencies of interest. Once the magnitude response in our experiment has been set, we need to perform group delay equalization to yield approximately constant group delay using a least-pth algorithm to constrain the radius of the poles. To construct a LPF, we employ a frequency response function of the Butterworth LPF. This type of filter is especially useful because the random errors involved in the raw position EEG data obtained through reconstruction are characterized by relatively high frequency contents. The Butterworth


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LPF can be expressed by 2

|Hc (jω)| =

1 , 1 + (jω/jωc )2n

(4.27)

√ where j = −1, ω is the frequency (rad/s), ωc is the cutoff frequency (rad/s), and n = the order of the filter. When ω = 0, the magnitude-squared function (Hc2 ) equals 1 and the frequency component is completely passed. When ω = ∞ , Hc2 equals 0 and the frequency component is completely stopped. Between the passband and the stopband, there is the transition band (0 < Hc2 < 1) in which the frequency component will be partially passed but partially stopped at the same time. When ω = ωc , Hc2 always equals 0.5 (half-power) regardless of the order of the filter. The Butterworth LPF is usually¡ represented by the transfer function of its normalized form, ¢ 2 |H(jω)| = 1/ 1 + ω 2n . The frequency response function of the Butterworth filter involves complex numbers (jω). Thus, the magnitude-squared function is the product of the response function pairs Hc (jω) and Hc (−jω) given by 2

|Hc (jω)| = Hc (jω) · Hc (−jω) =

1 . 1 + (jω/jωc )2n

The Butterworth LPF gives transfer functions that are rational functions but finding its roots results in a transfer function of the form: 1 1 = , n−1 (s − s1 )(s − s2 ) · · · (s − sn ) sn + an−1 s + · · · + a1 s + 1 where ai is the root of 1 + (−1)n s2n = 0.

4.3.2 Hypothesis II Before the ictal period, there are frequency components that have frequency responses that can be used to verify that the brain is fundamentally parallel. In this study, neuropsychological and neuroanatomical evidence might suggest that both its function and its organization are modular. To what extent can parallels be drawn between modularity of epileptogenesis processes as exemplified by cerebral localization of function which might be revealed by an epileptogenic zone? In addition, we speculate that this phenomenon might be a result of quadratic phase couplings. We hypothesize that this phenomenon is a result of quadratic phase couplings, which can be manifested through the bispectrum of the EEG signal by decomposing the Wigner—Ville time—frequency components of the EEG signal. The frequency components appearing in the spectrogram should form a nonperfect sine wave that creates a higher-order harmonic of the signals. By employing Choi—Williams time—frequency distribution, we should be able to capture the frequency responses of the EEG

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that show only the main component of EEG signals (not the harmonic component), which is speculated to be reflected from the epileptogenesis processes.

Wigner—Ville Distribution The Wigner—Ville distribution (WVD) is employed to capture the temporal development of epileptogenesis, which might be reflected from the timedependent variation in amplitude changes of each frequency band. The WVD is defined as follows: Z ∞ ³ 1 τ´ ∗³ τ ´ 2πif τ W (t, f ) = h t− dτ, (4.28) h t+ e 2π −∞ 2 2 where h(t) is the time series data [7]. In practice, we normally use the discrete analogue of the previous equation, which is represented by Wjk = PN/2 2πik /N . The WVD is the most suitable and =−N/2 h(j− /2) h(j+ /2) e promising time—frequency distribution map for our study because it satisfies a large number of desirable mathematical properties including: (a) energy conservation (the energy of h can be obtained by integrating the WVD of h all over the time—frequency), (b) marginal properties (the energy spectral density and the instantaneous power can be obtained as marginal distributions of W (t, f )), and (c) compatibility with filterings (the WVD expresses the fact that if a signal y is the convolution of x and z, the WVD of y is the time-convolution between the WVD of z and the WVD of x).

Choi—Williams Distribution The Choi—Williams distribution (CWD) expresses the EEG time series from the spectral density point of view. The CWD, which is a time—frequency distribution of Cohen’s class, introduces an exponential function of the time series (also called exponential distribution). The exponential kernel is used to control the cross-terms as represented in the generalized ambiguity function domain. It is also considered to be the Fourier transform of the time-indexed autocorrelation function K(t, τ ) estimated at a given time t. The CWD is given by Z ∞ 1 C(t, f ) = ejτ f K(t, τ ) dτ, (4.29) 2π −∞ where ∙ ¸ ³ (u − t)2 τ´ τ´ ∗³ 1 p ·exp − u + ·h u − ·h du, (4.30) K(t, τ ) = 4τ 2 /σ 2 2 4πτ 2 /σ −∞ Z

∞

where σ is a factor controlling the suppression of cross-terms and the frequency resolution. Note that C(t, f ) becomes the Wigner—Ville distribution


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when σ → ∞. It also satisfies the marginal conditions for all the values of σ. To give a reasonable accuracy to the estimate K(t, τ ), the range of time average should be increased for the large value of τ [6]. It is worth mentioning that the Choi—Williams transform is quite expensive in terms of computational complexity.

4.4 Results Based on the experimental design, the results of these hypothesis testings will give more insights in complex physical components in different frequency bands of EEG signals, which are essential features of neocortical dynamics at multiple scales of interaction with different frequency bands and dominance of collective interactions at most scales, and periodic boundary conditions. In addition, the time—frequency component decomposition may provide insights to the brain network’s mechanism by which local and regional circuits can continuously form and reform with different regions functionally disconnected from other tissue (a form of self-organization). In the neurodynamics theory, the switching between more local and more global operation in time—frequency components is governed by local and global control parameters speculated to change due to the influences of various neuromodulators in epileptogenic processes. The following two sections explain the results of hypothesis testings and resultant observations in this study.

4.4.1 Hypothesis I Figure 4.8 illustrates ST Lmax profiles of original EEG data and EEG data after passing a lowpass filter (LPF) of 10, 20, 30, and 50 Hz. Obviously, the brain activity at Delta and Theta bands (under 10 Hz) is more chaotic than the activity at higher frequency bands. However, during a seizure, the drop in the chaoticity at Delta and Theta bands is much less prominent than that of higher frequency bands. After a seizure, we observe an activity at the high Gamma band (over 50 Hz), which makes the EEG signal become less chaotic (more ordered) 10 minutes after a seizure. It is worth noting that there is no significant activity in the EEG data between 20—50 Hz, which considered to be in high Beta and low Gamma bands. This result verifies the theory used to categorize frequency bands in EEG data based on neuroanatomical connections in the brain at higher frequency bands. If cortical synchrony of EEG data at different frequency bands were an indicator of epileptogenesis processes, it could be expected that ongoing activity between 10—50 Hz bands would invoke more complex integration processes during the seizure, which is reflected by more pronounced synchronized activity.

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Fig. 4.8 ST Lmax profiles of EEG data from Electrode 1: Unfiltered and after passing LPF 10, 20, 30, 40, and 50 Hz.

Based on the implication of our previous observation, the ST Lmax profile of filtered (LPF-10) appears to contain characteristics different from those at higher frequency bands. Also we observed that the EEG activity at 20—50 Hz bands does not contain pronounced characteristics associated with epileptogenic processes. Figure 4.9 illustrates ST Lmax profiles of original EEG data and EEG data from Electrode 1 (LTD1) after passing lowpass filter of 10 and 50 Hz, or bandpass filter (BPF) of 10 and 20 Hz. Surprisingly, the EEG activity at the frequency range of 10—20 Hz is much less chaotic than that at other frequency bands and the EEG activity at the frequency range lower than 10 Hz is much more chaotic than that at other frequency bands. This implies that EEG activity at Alpha and low Beta bands is much less chaotic than EEG data at all frequency bands. In addition, EEG activity at the Delta and Theta bands is much more chaotic than EEG data at all frequency bands. This phenomenon can be explained by the fact that more and less complex manipulative activity is also correlated with different patterns of synchronized oscillatory brain activity in different frequency ranges and synchronized Delta—Theta activity can be recorded from the motor and somatosensory cortices of the brain. Furthermore, Delta activity is related with attention deficit disorder which could occur in patients with epilepsy as they sometimes lose their ability to focus or maintain attention. Figure 4.10 illustrates ST Lmax profiles of original EEG data and EEG data from Electrode


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Fig. 4.9 ST Lmax profiles of EEG data from Electrode 1 after passing LPF 10 Hz, BPF 10—20 Hz, and LPF 50 Hz.

Fig. 4.10 Electrode 6 frequency band using LPF 10 BPF10—20 LPF 50.

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Fig. 4.11 Sweep frequency and ST Lmax profile of EEG data.

6 (RTD2) after passing a lowpass filter of 10 and 50 Hz, or bandpass filter of 10 and 20 Hz. Note that Figure 4.10 shows consistent findings as illustrated in Figure 4.9, which confirms our hypothesis. We also observed a drop of ST Lmax profiles of original EEG about 20 minutes after the seizure. This drop cannot be observed if the EEG data are filtered at LPF 50 Hz, which is postulated to be the enhancement of gamma-band responses over 50 Hz, which might be attributable to the level of sensorimotor integration required by complex movements or activity of the brain to recover from seizure episodes. These results are also consistent with the ones confirmed by an MEG investigation of human brain responses during the brain recovery period. In addition, the complexity of the brain activity could also be critical for the gamma-band response to occur as EEG recordings normally are used to investigate changes of gamma-band activity associated with tasks such as verbal and visual—spatial problem solving. Although these results are consistent with the assumption that cognitive processes, that is, selective attention to sensorimotor integration, underlie stronger gamma-band responses, they might also arise from the complexity of the movement to be performed. If no differences occur in these bands, the changes of the brain activity during seizure episodes cannot be the cause of an effect visible in lower frequencies. Figure 4.11 illustrates the sweep frequency and ST Lmax profile of EEG data.

4.4.2 Hypothesis II An example of 5 second EEG data before a seizure onset analyzed in this study for hypothesis II and its spectrogram are illustrated in Figure 4.12. We postulate that before the ictal period, there are frequency components that demonstrate neuropsychological and neuroanatomical evidence of the modularity of the brain network. In addition, epileptogenic processes are believed to be a result of quadratic phase couplings in the epileptogenic zone. Figure 4.13


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Fig. 4.12 An example of 5 sec data of EEG before a seizure onset and its spectrogram.

illustrates the bispectrum of the pre-ictal EEG signal by decomposing the Wigner time—frequency components. As shown, the decomposed frequency components form an imperfect sine wave, which is a result of higher-order harmonic of the pre-ictal EEG signal. In Figure 4.14, we employ the Choi— Williams time—frequency distribution to capture the main component of the frequency responses of the pre-ictal EEG signal, which can reflect a formation of the epileptogenic process. In this study, we have demonstrated that epileptogenic processes incur specific changes in high-frequency brain responses when two time—frequency components are compared. As observed in Figures 4.13 and 4.14, the time— frequency components have very different perceptual complexity and a difference in evoked spectral responses, which could be a reflection of neuronal recruitment that triggers the epileptogenic process. On the cortical level, the epileptogenesis should lead to cell assembly ignition, which gamma-band responses to the post-ictal period should be stronger than responses to the other stages (see Figure 4.8).

4.5 Conclusions and Discussion of Future Research This research has given critical contributions to neocortical dynamics and the nonlinear dynamics methods to study the brain dynamics and epileptogenic processes, which contain almost none of these features to significant degree, and are likely to have a wide spectrum of applications to successful theories of neocortical function. We have also demonstrated that successful theories must either approximate interactions between neuronal masses at microscopic and macroscopic levels consistent with the scale of the experiment (e.g., electrode size and location) by using modern statistical methods and methods in chaos theory to express variables at experimentally interesting scales in terms of degrees of chaoticity over microscopic and macroscopic variables and their distribution functions at different scales. In the future,

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Fig. 4.13 Wigner—Ville time—frequency of 5 sec EEG data before a seizure onset.

Fig. 4.14 Choi—Williams time—frequency of 5 sec EEG data before a seizure onset.


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we also plan to expand our research to take into account more from genuine neocortical theory, which is based on real anatomy and physiology (as understood at the time) and contains no free (arbitrary) parameters. Although this might be an indication that cortical spatiotemporal responses change with the chaoticity of the EEG, such as continuity, there is still much research to be done to study how these responses relate to perception of complex forms. Nevertheless, we can broadly characterize the brain network in terms of the dynamics underlying EEG information processing. The brain network modules are directly mapped onto the spatiotemporal properties of the EEG, which are governed by attractor dynamics extending over time. In attractor networks, the changes in the dynamics of the brain network can be reflected from epileptogenic stimuli as sets of initial points corresponding to those stimuli converge onto an attractor. During the pre-ictal and ictal periods, the complete convergence of the brain dynamics onto an attractor is postulated to be such a sequence of the ideal network’s response to epileptogenesis, which will be an infinite transient whose path is governed by the input and the attractor structure of the brain network. The results from this study suggested that different EEG frequency bands operate in different functions and have different architectures. This separation of different EEG frequency bands by function and “anatomy” is worth study because because it may explain correspondences between deficits in the brain network’s performance when specific brain functioning is damaged, which might be attributed to the localization of epileptogenesis. In the future, in order to address issues such as epileptogenesis localization, it is crucially important to understand representation and processing in connectivity models of the brain network as a tool for understanding neuropathologies. We postulated that there are crucial connections during epileptogenic processes governed by the pyramidal cells in which the probability of a synapse occurring between two cells varies inversely with the distance between them independent of their spatial separation. In addition, those connections can be driven by the long-range apical system of the brain connectivity, coordinated in its projections between cortical areas in a space-independent fashion. If one confines the time scale over which interactions occur to be short, it might be possible to regard these long-range connections as forming a number of feedforward systems of epileptogenesis. Ideally, if one can project back any signals, in theory, it is possible that the process would take too long to arrive to modify the initial feedforward stimulus and the epileptogenic processes be intervened. In addition, one possibility is to regard spatially localized collections of cells in the neocortex as components of a system. To monitor the time scale interactions of the brain network, we assume some randomness in the strength of individual connections, therefore, the strength of connections between groups of cells is approximately symmetric over a short range on average. Although the probability of connection between individual cells is space-dependent, this does not imply multiple connections between pairs of nearby cells.

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As fundamentally unstable brain cell units, their behavior is subject to change independent of their interactions with other neuronal groups. Furthermore, if the group of neuronal interactions to a symmetrically connected attractor is conceivable, it will also be highly constrained in the time scale to operate effectively. Another possible assumption for epileptogenic localization is that neuronal networks with spatially localized connections behave as selective dynamic patterns, forming systems in parallel to reaction—diffusion systems of brain connectivity. If this assumption holds, then the time and space constants for the decay of activity depending on the relative strengths and distributions of excitatory and inhibitory neurons and the level of “background” neuronal potential action activity in the system become critical. For this reason, in the future, we need to find an explanation of how the brain functions associated with epileptogenesis may be separated by time scales rather than anatomy theory. One may explain neuropsychological dissociations of function as neurochemical systems operating over different time courses as anatomical epileptogenic localizations. Sometimes it may be appropriate to equate “lesions” of connectivity models with localized epileptogenic zones. Nevertheless, brain dysfunction may be attributable to epileptogenesis to remote or diffuse modulatory systems. It also requires greater understanding in drawing inferences from neuropsychology to support modular connectivity models of epileptogenesis and to explain the basis of neuropathologies of epileptogenic zones. The studies of the connections of neocortical dynamics and neuronal networks also need to be addressed in order for us to have greater understanding about the brain. Our group has done some preliminary studies focusing on the idea that connections between assumed functional units at different scales (e.g., neurons, minicolumns, macrocolumns) can be symmetric or asymmetric, depending on the spatial scale of such units [30]. In a realm of network connectivity studies, the nature of connections at different scales is a critical theoretical issue. Also, features other than symmetry of such interactions (e.g., the density of the network’s connections) are important and addressed in that study [30]. Because neocortical dynamic variables may behave quite differently at different scales, the “interconnectivities” are studied. Furthermore, interactions across scales are also important, as emphasized in [30]. Any theory of EEG constructed at smaller scales must be coarse grained before comparisons are made with scalp data. In the future, we plan to experiment on the same analyses with some of these ideas on scalp EEG data. We expect chaotic or quasiperiodic behavior may be observed depending on the spatial filter implicit in the experimental methods. Acknowledgments The authors would like to thank members of the Bioengineering Research Partnership at the brain dynamics laboratory, Brain Institute, University of Florida, for their fruitful comments and discussion. Research was partially supported by Rutgers Research Council Grant-202018 and NIH grant R01-NS-39687-01A1.


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Chapter 5

Nonconvex Optimization for Communication Networks Mung Chiang

Summary. Nonlinear convex optimization has provided both an insightful modeling language and a powerful solution tool to the analysis and design of communication systems over the last decade. A main challenge today is on nonconvex problems in these applications. This chapter presents an overview on some of the important nonconvex optimization problems in communication networks. Four typical applications are covered: Internet congestion control through nonconcave network utility maximization, wireless network power control through geometric and sigmoidal programming, DSL spectrum management through distributed nonconvex optimization, and Internet intradomain routing through nonconvex, nonsmooth optimization. A variety of nonconvex optimization techniques are showcased: sum-of-squares programming through successive SDP relaxation, signomial programming through successive GP relaxation, leveraging specific structures in these engineering problems for efficient and distributed heuristics, and changing the underlying protocol to enable a different problem formulation in the first place. Collectively, they illustrate three alternatives of tackling nonconvex optimization for communication networks: going “through” nonconvexity, “around” nonconvexity, and “above” nonconvexity. Key words: Digital subscriber line, duality, geometric programming, Internet, network utility maximization, nonconvex optimization, power control, routing, semidefinite programming, sum of squares, TCP/IP, wireless network

Mung Chiang Electrical Engineering Department, Princeton University, Princeton, NJ 08544, U.S.A. e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_5, © Springer Science+Business Media, LLC 2009

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5.1 Introduction There have been two major “waves” in the history of optimization theory and its applications: the first started with linear programming (LP) and the simplex method in the late 1940s, and the second with convex optimization and the interior point method in the late 1980s. Each has been followed by a transforming period of “appreciation-application cycle”: as more people appreciate the use of LP/convex optimization, more look for their formulations in various applications; then more work on its theory, efficient algorithms, and software; the more powerful the tools become, and in turn more people appreciate its usage. Communication systems benefit significantly from both waves; the vast array of many success stories includes multicommodity flow solutions (e.g., Bellman—Ford algorithm) from LP, and network utility maximization and robust transceiver design from convex optimization. Much of the current research is about the potential of the third wave, on nonconvex optimization. If one word is used to differentiate between easy and hard problems, convexity is probably the “watershed.” But if a longer description length is allowed, useful conclusions can be drawn even for nonconvex optimization. Indeed, convexity is a very disturbing watershed, because it is not a topological invariant under change of variable (e.g., see geometric programming) or higher-dimension embedding (e.g., see sum of squares method). A variety of approaches has been proposed to tackle nonconvex optimization problems: from successive convex approximation to dualization, from nonlinear transformation to turn an apparently nonconvex problem into a convex problem to characterization of attraction regions and systematically jumping out of a local optimum, and from leveraging the specific structures of the problems (e.g., difference of convex functions, concave minimization, low rank nonconvexity) to developing more efficient branch-and-bound procedures. Researchers in communications and networking have been examining nonconvex optimization using domain-specific structures in important problems in the areas of wireless networking, Internet engineering, and communication theory. Perhaps four typical topics best illustrate the variety of challenging issues arising from nonconvex optimization in communication systems: • Nonconvex objective to be minimized. An example is congestion control for inelastic application traffic, where a nonconcave utility function needs to be maximized. • Nonconvex constraint set. An example is power control in the low SIR regime. • Integer constraints. Two important examples are single path routing and multiuser detection. • Constraint sets that are convex but require an exponential number of inequalities to explicitly describe. An example is optimal scheduling in multihop wireless networks under certain interference models. The problem of wireless scheduling will not be discussed in this chapter. Interested readers can refer to [73] for a unifying framework of the problem.

5 Nonconvex Optimization for Communication Networks

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This chapter overviews the latest results in recent publications about the first two topics, with a particular focus on showing the connections between the engineering intuitions about important problems in communication networks and the state-of-the-art algorithms in nonconvex optimization theory. Most of the results surveyed here were obtained in 2005—2006, and the problems driven by fundamental issues in the Internet, wireless, and broadband access networks. As this chapter illustrates, even after much progress made in recent years, there are still many challenging mysteries to be resolved on these important nonconvex optimization problems.

1

2

3

Fig. 5.1 Three major types of approaches when tackling nonconvex optimization problems in communication networks: Go (1) through, (2) around, or (3) above nonconvexity.

It is interesting to point out that, as illustrated in Figure 5.1, there are at least three very different approaches to tackle the difficult issue of nonconvexity. • Go “through” nonconvexity. In this approach, we try to solve the difficult nonconvex problem; for example, we may use successive convex relaxations (e.g., sum-of-squares, signomial programming), utilize special structures in the problem (e.g., difference of convex functions, generalized quasiconcavity), or leverage smarter branch and bound methods. • Go “around” nonconvexity. In this approach, we try to avoid solving the convex problem; for example, we may discover a change of variables that turns the seemingly nonconvex problem into a convex one, determine conditions under which the problem is convex or the KKT point is unique, or make approximations to make the problem convex. • Go “above” nonconvexity. In this approach, we try to reformulate the nonconvex problem in the first place to make it more “solvable” or “approximately solvable.” We observe that optimization problem formulations are induced by some underlying assumptions on what the network architectures and protocols should look like. By changing these assumptions, a different, much easier-to-solve or easier-to-approximate formulations may result. We refer to this approach as design for optimizability, which is concerned with redrawing architectures to make the resulting optimization

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problem easier to solve. This approach of changing a hard problem into an easier one is in contrast to optimization, which tries to solve a given, possibly difficult, problem. The four topics chosen in this chapter span a range of application contexts and tasks in communication networks. The sources of difficulty in these nonconvex optimization problems are summarized in Table 5.1, together with the key ideas in solving them and the type of approaches used. For more details beyond this brief overview chapter, please refer to the related publications [29, 19, 14, 35, 7, 70, 71] by the author and coworkers and the references therein. Table 5.1 Summary of four nonconvex optimization problems in this chapter Section Application 5.2 Internet 5.3

Wireless

5.4

DSL

5.5

Internet

Task Difficulty Solution Approach Congestion Nonconcave U Sum of “Through” control squares Power Posynomial Geometric “Around” control ratio program Spectrum Posynomial Problem “Around” management ratio structure Routing Nonconvex Approximation “Above” constraint

5.2 Internet Congestion Control 5.2.1 Introduction Basic Network Utility Maximization Since the publication of the seminal paper [37] by Kelly, Maulloo, and Tan in 1998, the framework of network utility maximization (NUM) has found many applications in network rate allocation algorithms and Internet congestion control protocols (e.g., surveyed in [45, 60]). It has also led to a systematic understanding of the entire network protocol stack in the unifying framework of “layering as optimization decomposition” (e.g., surveyed in [13, 49, 44]). By allowing nonlinear concave utility objective functions, NUM substantially expands the scope of the classical LP-based network flow problems. Consider a communication network with L links, each with a fixed capacity of cl bps, and S sources (i.e., end-users), each transmitting at a source rate of xs bps. Each source s emits one flow, using a fixed set L(s) of links in its path, and has a utility function Us (xs ). Each link l is shared by a set S(l) of


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sources. Network utility maximization, in its basic version, P is the following problem of maximizing the total utility of the network s Us (xs ), over the P source rates x, subject to linear flow constraints s:l∈L(s) xs ≤ cl for all links l: maximize subject to

P

s

P

Us (xs )

s∈S(l)

xs ≤ cl , ∀l,

(5.1)

x º 0,

where the variables are x ∈ RS . There are many nice properties of the basic NUM model due to several simplifying assumptions of the utility functions and flow constraints, which provide the mathematical tractability of problem (5.1) but also limit its applicability. In particular, the utility functions {Us } are often assumed to be increasing and strictly concave functions. Assuming that Us (xs ) becomes concave for large enough xs is reasonable, because the law of diminishing marginal utility eventually will be effective. However, Us may not be concave throughout its domain. In his seminal paper in 1995, Shenker [57] differentiated inelastic network traffic from elastic traffic. Utility functions for elastic traffic were modeled as strictly concave functions. Although inelastic flows with nonconcave utility functions represent important applications in practice, they have received little attention and rate allocation among them has only a limited mathematical foundation. There have been three recent publications [41, 29, 19] (see also earlier work in [69, 42, 43] related to the approach in [41]) on this topic. In this section, we investigate the extension of the basic NUM to maximization of nonconcave utilities, as in the approach of [19]. We provide a centralized algorithm for offline analysis and establishment of a performance benchmark for nonconcave utility maximization when the utility function is a polynomial or signomial. Based on the semialgebraic approach to polynomial optimization, we employ convex sum-of-squares (SOS) relaxations solved by a sequence of semidefinite programs (SDP), to obtain increasingly tighter upper bounds on total achievable utility for polynomial utilities. Surprisingly, in all our experiments, a very low-order and often a minimal-order relaxation yields not just a bound on attainable network utility, but the globally maximized network utility. When the bound is exact, which can be proved using a sufficient test, we can also recover a globally optimal rate allocation.

Canonical Distributed Algorithm A reason that the assumption of a utility function’s concavity is upheld in many papers on NUM is that it leads to three highly desirable mathematical properties of the basic NUM:

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• It is a convex optimization problem, therefore the global minimum can be computed (at least in centralized algorithms) in worst-case polynomialtime complexity [4]. • Strong duality holds for (5.1) and its Lagrange dual problem. A zero duality gap enables a dual approach to solve (5.1). • Minimization of a separable objective function over linear constraints can be conducted by distributed algorithms based on the dual approach. Indeed, the basic NUM (5.1) is such a “nice” optimization problem that its theoretical and computational properties have been well studied since the 1960s in the field of monotropic programming (e.g., as summarized in [54]). For network rate allocation problems, a dual-decomposition-based distributed algorithm has been widely studied (e.g., in [37, 45]), and is summarized below. Zero duality gap for (5.1) states that solving the Lagrange dual problem is equivalent to solving the primal problem (5.1). The Lagrange dual problem is readily derived. We first form the Lagrangian of (5.1): ⎛ ⎞ X X X L(x, λ) = Us (xs ) + λl ⎝cl − xs ⎠ , s

l

s∈S(l)

where λl ≥ 0 is the Lagrange multiplier (can be interpreted as the link congestion price) associated with the linear flow constraint on link l. Additivity of total utility and linearity of flow constraints lead to a Lagrangian dual decomposition into individual source terms: ⎡ ⎛ ⎞ ⎤ X X X ⎣Us (xs ) − ⎝ λl ⎠ xs ⎦ + cl λl L(x, λ) = s

=

X

l∈L(s)

s

Ls (xs , λ ) +

s

X

l

cl λl ,

l

P where λs = l∈L(s) λl . For each source s, Ls (xs , λs ) = Us (xs ) − λs xs only depends on local xs and the link prices λl on those links used by source s. The Lagrange dual function g(λ) is defined as the maximized L(x, λ) over x. This “net utility” maximization obviously can be conducted distributively P by each source, as long as the aggregate link price λs = l∈L(s) λl is available to source s, where source s maximizes a strictly concave function Ls (xs , λs ) over xs for a given λs : x∗s (λs ) = argmax [Us (xs ) − λs xs ] , ∀s.

(5.2)

The Lagrange dual problem is minimize g(λ) = L(x∗ (λ), λ) subject to λ º 0,

(5.3)


143

where the optimization variable is λ. Any algorithms that find a pair of primal—dual variables (x, λ) that satisfy the KKT optimality condition would solve (5.1) and its dual problem (5.3). One possibility is a distributed, iterative subgradient method, which updates the dual variables λ to solve the dual problem (5.3): ⎡

⎛

λl (t + 1) = ⎣λl (t) − α(t) ⎝cl −

X

s∈S(l)

⎞⎤+

xs (λs (t))⎠⎦ , ∀l,

(5.4)

where t is the iteration number and α(t) > 0 are step sizes. Certain choices of step sizes, such as α(t) = α0 /t, α0 > 0, guarantee that the sequence of dual variables λ(t) will converge to the dual optimal λ∗ as t → ∞. The primal variable x(λ(t)) will also converge to the primal optimal variable x∗ . For a primal problem that is a convex optimization, the convergence is towards the global optimum. The sequence of the pair of algorithmic steps (5.2, 5.4) forms what we refer to as the canonical distributed algorithm, which solves the network utility optimization problem (5.1) and the dual (5.3) and computes the optimal rates x∗ and link prices λ∗ .

Nonconcave Network Utility Maximization It is known that for many multimedia applications, user satisfaction may assume a nonconcave shape as a function of the allocated rate. For example, the utility for voice applications is better described by a sigmoidal function: with a convex part at low rate and a concave part at high rate, and a single inflexion point x0 (with Us00 (x0 ) = 0) separating the two parts. Furthermore, in some other models of utility functions, the concavity assumption on Us is also related to the elasticity assumption on rate demands by users. When demands for xs are not perfectly elastic, Us (xs ) may not be concave. Suppose we remove the critical assumption that {Us } are concave functions, and allow them to be any nonlinear functions. The resulting NUM becomes nonconvex optimization and significantly harder to be analyzed and solved, even by centralized computational methods. In particular, a local optimum may not be a global optimum and the duality gap can be strictly positive. The standard distributive algorithms that solve the dual problem may produce infeasible or suboptimal rate allocation. There have been several recent publications on distributed algorithms for nonconcave utility maximization. In [41], a “self-regulation” heuristic is proposed to avoid the resulting oscillation in rate allocation and is shown to converge to an optimal rate allocation asymptotically when the proportion of nonconcave utility sources vanishes. In [29], a set of sufficient conditions and necessary conditions is presented under which the canonical distributed algo-

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2.5

U(x)

2

1.5

1

0.5

0

0

2

4

6

x

8

10

12

Fig. 5.2 Some examples of utility functions Us (xs ): it can be concave or sigmoidal as shown in the graph, or any general nonconcave function. If the bottleneck link capacity used by the source is small enough, that is, if the dotted vertical line is pushed to the left, a sigmoidal utility function effectively becomes a convex utility function.

rithm still converges to the globally optimal solution. However, these conditions may not hold in many cases. These two approaches illustrate the choice between admission control and capacity planning to deal with nonconvexity (see also the discussion in [36]). But neither approach provides a theoretically polynomial-time and practically efficient algorithm (distributed or centralized) for nonconcave utility maximization. In [19], using a family of convex semidefinite programming (SDP) relaxations based on the sum-of-squares (SOS) relaxation and the positivstellensatz theorem in real algebraic geometry, we apply a centralized computational method to bound the total network utility in polynomial time. A surprising result is that for all the examples we have tried, wherever we could verify the result, the tightest possible bound (i.e., the globally optimal solution) of NUM with nonconcave utilities is computed with a very low-order relaxation. This efficient numerical method for offline analysis also provides the benchmark for distributed heuristics. These three different approaches: proposing distributed but suboptimal heuristics (for sigmoidal utilities) in [41], determining optimality conditions for the canonical distributed algorithm to converge globally (for all nonlinear utilities) in [29], and proposing an efficient but centralized method to compute the global optimum (for a wide class of utilities that can be transformed into polynomial utilities) in [19] (and this section), are complementary in the study of distributed rate allocation by nonconcave NUM.


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5.2.2 Global Maximization of Nonconcave Network Utility Sum-of-Squares Method We would like to bound the maximum network utility by γ in polynomial time and search for a tight bound. Had there been no link capacity constraints, maximizing a polynomial is already an NP-hard problem, but can be relaxed into an SDP [58]. This is because testing if the following bounding inequality holds γ ≥ p(x), where p(x) is a polynomial of degree d in n variables, is equivalent to testing the positivity of γ − p(x), which can be relaxed P into testing if γ − p(x) can be written as a sum of squares (SOS): p(x) = ri=1 qi (x)2 for some polynomials qi , where the degree of qi is less than or equal to d/2. This is referred to as the SOS relaxation. If a polynomial can be written as a sum of squares, it must be nonnegative, but not vice versa. Conditions under which this relaxation is tight have been studied since Hilbert. Determining if a sum of squares decomposition exists can be formulated as an SDP feasibility problem, thus polynomial-time solvable. Constrained nonconcave NUM can be relaxed by a generalization of the Lagrange duality theory, which involves nonlinear combinations of the constraints instead of linear combinations in the standard duality theory. The key result is the positivstellensatz, due to Stengle [62], in real algebraic geometry, which states that for a system of polynomial inequalities, either there exists a solution in Rn or there exists a polynomial which is a certificate that no solution exists. This infeasibility certificate has recently been shown to be also computable by an SDP of sufficient size [51, 50], a process that is referred to as the sum-of-squares method and automated by the software SOSTOOLS [52] initiated by Parrilo in 2000. For a complete theory and many applications of SOS methods, see [51] and references therein. Furthermore, the bound γ itself can become an optimization variable in the SDP and can be directly minimized. A nested family of SDP relaxations, each indexed by the degree of the certificate polynomial, is guaranteed to produce the exact global maximum. Of course, given the problem is NP-hard, it is not surprising that the worst-case degree of certificate (thus the number of SDP relaxations needed) is exponential in the number of variables. What is interesting is the observation that in applying SOSTOOLS to nonconcave utility maximization, a very low-order, often the minimum-order relaxation already produces the globally optimal solution.

Application of SOS Method to Nonconcave NUM Using sum-of-squares and the positivstellensatz, we set up the following problem whose objective value converges to the optimal value of problem (5.1),

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where {Ui } are now general polynomials, as the degree of the polynomials involved is increased. minimize γ subject P to P P γ − s Us (xs ) − l λl (x)(cl − s∈S(l) xs ) P P P − j,k λjk (x)(cj − s∈S(j) xs )(ck − s∈S(k) xs )− P P . . . − λ12...n (x)(c1 − s∈S(1) xs ) . . . (cn − s∈S(n) xs ) is SOS, λl (x), λjk (x), . . . , λ12...n (x) are SOS.

(5.5)

The optimization variables are γ and all of the coefficients in polynomials λl (x), λjk (x), . . ., λ12...n (x). Note that x is not an optimization variable; the constraints hold for all x, therefore imposing constraints on the coefficients. This formulation uses Schm¨ udgen’s representation of positive polynomials over compact sets [56]. Let D be the degree of the expression in the first constraint in (5.5). We refer to problem (5.5) as the SOS relaxation of order D for the constrained NUM. For a fixed D, the problem can be solved via SDP. As D is increased, the expression includes more terms, the corresponding SDP becomes larger, and the relaxation gives tighter bounds. An important property of this nested family of relaxations is guaranteed convergence of the bound to the global maximum. Regarding the choice of degree D for each level of relaxation, clearly a polynomial of odd degree cannot be SOS, so we need to consider only the cases where the expression has even degree. Therefore, the degree of the first nontrivial P relaxation is the largest even number greater than or equal to degree s Us (xs ), and the degree is increased by 2 for the next level. A key question now becomes: how do we find out, after solving an SOS relaxation, if the bound happens to be exact? Fortunately, there is a sufficient test that can reveal this, using the properties of the SDP and its dual solution. In [31, 39], a parallel set of relaxations, equivalent to the SOS ones, is developed in the dual framework. The dual of checking the nonnegativity of a polynomial over a semialgebraic set turns out to be finding a sequence of moments that represent a probability measure with support in that set. To be a valid set of moments, the sequence should form a positive semidefinite moment matrix. Then, each level of relaxation fixes the size of this matrix (i.e., considers moments up a certain order) and therefore solves an SDP. This is equivalent to fixing the order of the polynomials appearing in SOS relaxations. The sufficient rank test checks a rank condition on this moment matrix and recovers (one or several) optimal x∗ , as discussed in [31]. In summary, we have the following algorithm for centralized computation of a globally optimal rate allocation to nonconcave utility maximization, where the utility functions can be written as or converted into polynomials.


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Algorithm 1. Sum-of-squares for nonconcave utility maximization. 1. Formulate the relaxed problem (5.5) for a given degree D. 2. Use SDP to solve the Dth order relaxation, which can be conducted using SOSTOOLS [52]. 3. If the resulting dual SDP solution satisfies the sufficient rank condition, the Dth-order optimizer γ ∗ (D) is the globally optimal network utility, and a corresponding x∗ can be obtained.1 4. Increase D to D+2, that is, the next higher-order relaxation, and repeat. In the following section, we give examples of the application of SOS relaxation to the nonconcave NUM. We also apply the above sufficient test to check if the bound is exact, and if so, we recover the optimum rate allocation x∗ that achieve this tightest bound.

5.2.3 Numerical Examples and Sigmoidal Utilities Polynomial Utility Examples First, consider quadratic utilities (i.e., Us (xs ) = x2s ) as a simple case to start with (this can be useful, for example, when the bottleneck link capacity limits sources to their convex region of a sigmoidal utility). We present examples that are typical, in our experience, of the performance of the relaxations. x2

x3

c1

c2

x1 Fig. 5.3 Network topology for Example 5.1.

Example 5.1. A small illustrative example. Consider the simple 2-link, 3user network shown in Figure 5.3, with c = [1, 2]. The optimization problem is P 2 maximize s xs subject to x1 + x2 ≤ 1 (5.6) x1 + x3 ≤ 2 x1 , x2 , x3 ≥ 0. 1 Otherwise, γ ∗ (D) may still be the globally optimal network utility but is only provably an upper bound.

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The first level relaxation with D = 2 is minimize γ subject to γ − (x21 + x22 + x23 ) − λ1 (−x1 − x2 + 1) − λ2 (−x1 −x3 + 2) − λ3 x1 − λ4 x2 − λ5 x3 − λ6 (−x1 − x2 + 1) (−x1 − x3 + 2) − λ7 x1 (−x1 − x2 + 1) − λ8 x2 (−x1 −x2 + 1) − λ9 x3 (−x1 − x2 + 1) − λ10 x1 (−x1 − x3 + 2) −λ11 x2 (−x1 − x3 + 2) − λ12 x3 (−x1 − x3 + 2)− λ13 x1 x2 − λ14 x1 x3 − λ15 x2 x3 is SOS, λi ≥ 0, i = 1, . . . , 15.

(5.7)

The first constraint above can be written as xT Qx for x = [1, x1 , x2 , x3 ]T and an appropriate Q. For example, the (1,1) entry which is the constant term reads γ − λ1 − 2λ2 − 2λ6 , the (2,1) entry, coefficient of x1 , reads λ1 + λ2 − λ3 + 3λ6 − λ7 − 2λ10 , and so on. The expression is SOS if and only if Q ≥ 0. The optimal γ is 5, which is achieved by, for example, λ1 = 1, λ2 = 2, λ3 = 1, λ8 = 1, λ10 = 1, λ12 = 1, λ13 = 1, λ14 = 2 and the rest of the λi equal to zero. Using the sufficient test (or, in this example, by inspection) we find the optimal rates x0 = [0, 1, 2]. In this example, many of the λi could be chosen to be zero. This means not all product terms appearing in (5.7) are needed in constructing the SOS polynomial. Such information is valuable from the decentralization point of view, and can help determine to what extent our bound can be calculated in a distributed manner. This is a challenging topic for future work.

c3

c1

c2

c6 c5

c4 c7

Fig. 5.4 Network topology for Example 5.2.

Example 5.2. Larger tree topology. As a larger example, consider the network shown in Figure 5.4 with seven links. There are nine users, with the following routing table that lists the links on each user’s path. x1 x2 x3 x4 x5 x6 x7 x8 x9 1,2 1,2,4 2,3 4,5 2,4 6,5,7 5,6 7 5


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For c = [5, 10, 4, 3, 7, 3, 5], we obtain the bound γ = 116 with D = 2, which turns out to be globally optimal, and the globally optimal rate vector can be recovered: x0 = [5, 0, 4, 0, 1, 0, 0, 5, 7]. In this example, exhaustive search is too computationally intensive, and the sufficient condition test plays an important role in proving the bound is exact and in recovering x0 .

c1 c6

c2

c5

c3 c4

Fig. 5.5 Network topology for Example 5.3.

Example 5.3. Large m-hop ring topology. Consider a ring network with n nodes, n users, and n links where each user’s flow starts from a node and goes clockwise through the next m links, as shown in Figure 5.5 for n = 6, m = 2. As a large example, with n = 25, m = 2, and capacities chosen randomly for a uniform distribution on [0, 10], using relaxation of order D = 2 we obtain the exact bound γ = 321.11 and recover an optimal rate allocation. For n = 30, m = 2, and capacities randomly chosen from [0, 15], it turns out that D = 2 relaxation yields the exact bound 816.95 and a globally optimal rate allocation.

Sigmoidal Utility Examples Now consider sigmoidal utilities in a standard form: Us (xs ) =

1 1+

e−(as xs +bs )

,

where {as , bs } are constant integers. Even though these sigmoidal functions are not polynomials, we show the problem can be cast as one with polynomial cost and constraints, with a change of variables. Example 5.4. Sigmoidal utility. Consider the simple 2-link, 3-user example shown in Figure 5.3 for as = 1 and bs = −5. The NUM problem is to

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P

1 maximize s 1+e−(xs −5) subject to x1 + x2 ≤ c1 x1 + x3 ≤ c2 x ≥ 0.

(5.8)

¡ ¢ Let ys = 1/ 1 + e−(xs −5) , then xs = − log((1/ys ) − 1) + 5. Substituting for x1 , x2 in the first constraint, arranging terms and taking exponentials, then multiplying the sides by y1 y2 (note that y1 , y2 > 0), we get (1 − y1 )(1 − y2 ) ≥ e(10−c1 ) y1 y2 , which is polynomial in the new variables y. This applies to all capacity¡constraints, and the nonnegativity constraints for xs translate to ys ≥ ¢ 1/ 1 + e5 . Therefore the whole problem can be written in polynomial form, and SOS methods apply. This transformation renders the problem polynomial for general sigmoidal utility functions, with any as and bs . We present some numerical results, using a small illustrative example. Here SOS relaxations of order 4 (D = 4) were used. For c1 = 4, c2 = 8, we find γ = 1.228, which turns out to be a global optimum, with x0 = [0, 4, 8] as the optimal rate vector. For c1 = 9, c2 = 10, we find γ = 1.982 and x0 = [0, 9, 10]. Now place a weight of 2 on y1 , and the other ys have weight one; we obtain γ = 1.982 and x0 = [9, 0, 1]. In general, if as 6= 1 for some s, however, the degree of the polynomials in the transformed problem may be very high. If we write the general problem as P 1 maximize Ps 1+e−(as xs +bs ) subject to s∈S(l) xs ≤ cl , ∀l, (5.9) x ≥ 0, each capacity constraint after transformation will be Q rls Πk6=s ak ≥ s (1 − ys ) Q P Q rls Q k6=s ak exp(− s as (cl + s rls /as bs )) s ys ,

where rls = 1 if l ∈ L(s) and equals 0 otherwise. Because the product of the as appears in the exponents, as > 1 significantly increases the degree of the polynomials appearing in the problem and hence the dimension of the SDP in the SOS method. It is therefore also useful to consider alternative representations of sigmoidal functions such as the following rational function: Us (xs ) =

xns , a + xns

where the inflection point is x0 = ((a(n − 1)) / (n + 1))1/n and the slope at the inflection point is Us (x0 ) = ((n − 1) /4n) ((n + 1) / (a(n − 1)))1/n . Let


ys = Us (xs ); the NUM problem in this case is equivalent to P maximize s ys n n subject to x Ps − ys xs − ays = 0 s∈S(l) xs ≤ cl , ∀l x≥0

151

(5.10)

which again can be accommodated in the SOS method and be solved by Algorithm 1. The benefit of this choice of utility function is that the largest degree of the polynomials in the problem is n + 1, therefore growing linearly with n. The disadvantage compared to the exponential form for sigmoidal functions is that the location of the inflection point and the slope at that point cannot be set independently.

5.2.4 Alternative Representations for Convex Relaxations to Nonconcave NUM The SOS relaxation we used in the last two sections is based on Schm¨ udgen’s representation for positive polynomials over compact sets described by other polynomials. We now briefly discuss two other representations of relevance to the NUM, that are interesting from both theoretical (e.g., interpretation) and computational points of view.

LP Relaxation Exploiting linearity of the constraints in NUM and with the additional assumption of nonempty interior for the feasible set (which holds for NUM), we can use Handelman’s representation [30] and refine the positivstellensatz condition to obtain the following convex relaxation of nonconcave NUM problem. maximize γ subject to L X Y P P γ − s Us (xs ) = λα (cl − s∈S(l) xs )αl , ∀x α∈N L

(5.11)

l=1

λα ≥ 0, ∀α,

where the optimization variables are γ and λα , and α denotes an ordered set of integers {αl }. P Fixing D where l αl ≤ D, and equating the coefficients on the two sides of the equality in (5.11), yields a linear program (LP). There are no

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SOS terms, therefore no semidefiniteness conditions. As before, increasing the degree D gives higher-order relaxations and a tighter bound. We provide a pricing interpretation for problem (5.11).PFirst, normalize each capacity constraint as 1 − ul (x) ≥ 0, where ul (x) = s∈S(l) xs /cl . We can interpret ul (x) as link usage, or the probability that link l is used at any given point in time. Then, in (5.11), we have terms linear in u such as λl (1 − ul (x)), in which λl has a similar interpretation as in concave NUM, as the price of using link l. We also have product terms such as λjk (1−uj (x))(1− uk (x)), where λjk uj (x)uk (x) indicates the probability of simultaneous usage of links j and k, for links whose usage probabilities are independent (e.g., they do not share any flows). Products of more terms can be interpreted similarly. Although the above price interpretation is not complete and does not justify all the terms appearing in (5.11) (e.g., powers of the constraints, product terms for links with shared flows), it does provide some useful intuition: this relaxation results in a pricing scheme that provides better incentives for the users to observe the constraints, by giving an additional reward (because the corresponding term adds positively to the utility) for simultaneously keeping two links free. Such incentive helps tighten the upper bound and eventually achieve a feasible (and optimal) allocation. This relaxation is computationally attractive because we need to solve an LPs instead of the previous SDPs at each level. However, significantly more levels may be required [40].

Relaxation with No Product Terms Putinar [53] showed that a polynomial positive over a compact set2 can be represented as an SOS-combination of the constraints. This yields the following convex relaxation for nonconcave NUM problem. maximize γ subject to P P P γ − s Us (xs ) = L s∈S(l) xs ), ∀x l=1 λl (x)(cl − λ(x) is SOS,

(5.12)

where the optimization variables are the coefficients in λl (x). Similar to the SOS relaxation (5.5), fixing the order D of the expression in (5.12) results in an SDP. This relaxation has the nice property that no product terms appear: the relaxation becomes exact with a high enough D without the need of product terms. However, this degree might be much higher than what the previous SOS method requires.

2

With an extra assumption that always holds for linear constraints as in NUM problems.


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5.2.5 Concluding Remarks and Future Directions We consider the NUM problem in the presence of inelastic flows, that is, flows with nonconcave utilities. Despite its practical importance, this problem has not been studied widely, mainly due to the fact it is a nonconvex problem. There has been no effective mechanism, centralized or distributed, to compute the globally optimal rate allocation for nonconcave utility maximization problems in networks. This limitation has made performance assessment and design of networks that include inelastic flows very difficult. In one of the recent works on this topic [19], we employed convex SOS relaxations, solved by a sequence of SDPs, to obtain high-quality, increasingly tighter upper bounds on total achievable utility. In practice, the performance of our SOSTOOLS-based algorithm was surprisingly good, and bounds obtained using a polynomial-time (and indeed a low-order and often minimalorder) relaxation were found to be exact, achieving the global optimum of nonconcave NUM problems. Furthermore, a dual-based sufficient test, if successful, detects the exactness of the bound, in which case the optimal rate allocation can also be recovered. This performance of the proposed algorithm brings up a fundamental question on whether there is any particular property or structure in nonconcave NUM that makes it especially suitable for SOS relaxations. We further examined the use of two more specialized polynomial representations, one that uses products of constraints with constant multipliers, resulting in LP relaxations; and at the other end of spectrum, one that uses a linear combination of constraints with SOS multipliers. We expect these relaxations to give higher-order certificates, thus their potential computational benefits need to be examined further. We also show they admit economics interpretations (e.g., prices, incentives) that provide some insight on how the SOS relaxations work in the framework of link congestion pricing for the simultaneous usage of multiple links. An important research issue to be further investigated is decentralization methods for rate allocation among sources with nonconcave utilities. The proposed algorithm here is not easy to decentralize, given the products of the constraints or polynomial multipliers that destroy the separable structure of the problem. However, when relaxations become exact, the sparsity pattern of the coefficients can provide information about partially decentralized computation of optimal rates. For example, if after solving the NUM offline, we obtain an exact bound, then if the coefficient of the cross-term xi xj turns out to be zero, it means users i and j do not need to communicate to each other to find their optimal rates. An interesting next step in this area of research is to investigate a distributed version of the proposed algorithm through limited message passing among clusters of network nodes and links.

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5.3 Wireless Network Power Control 5.3.1 Introduction Due to the broadcast nature of radio transmission, data rates and other quality of service (QoS) issues in a wireless network are affected by interference. This is particularly important in CDMA systems where users transmit at the same time over the same frequency bands and their spreading codes are not perfectly orthogonal. Transmit power control is often used to tackle this problem of signal interference [12]. We study how to optimize over the transmit powers to create the optimal set of signal-to-interference ratios (SIR) on wireless links. Optimality here can be with respect to a variety of objectives, such as maximizing a systemwide efficiency metric (e.g., the total system throughput), or maximizing a QoS metric for a user in the highest QoS class, or maximizing a QoS metric for the user with the minimum QoS metric value (i.e., a maxmin optimization). The objective represents a systemwide goal to be optimized; however, individual users’ QoS requirements also need to be satisfied. Any power allocation must therefore be constrained by a feasible set formed by these minimum requirements from the users. Such a constrained optimization captures the tradeoff between user-centric constraints and some network-centric objective. Because a higher power level from one transmitter increases the interference levels at other receivers, there may not be any feasible power allocation to satisfy the requirements from all the users. Sometimes an existing set of requirements can be satisfied, but when a new user is admitted into the system, there exist no more feasible power control solutions, or the maximized objective is reduced due to the tightening of the constraint set, leading to the need for admission control and admission pricing, respectively. Because many QoS metrics are nonlinear functions of SIR, which is in turn a nonlinear (and neither convex nor concave) function of transmit powers, in general, power control optimization or feasibility problems are difficult nonlinear optimization problems that may appear to be NP-hard problems. Following [14, 35], this section shows that, when SIR is much larger than 0 dB, a class of nonlinear optimization called geometric programming (GP) can be used to efficiently compute the globally optimal power control in many of these problems, and efficiently determine the feasibility of user requirements by returning either a feasible (and indeed optimal) set of powers or a certificate of infeasibility. This also leads to an effective admission control and admission pricing method. The key observation is that despite the apparent nonconvexity, through log change of variable the GP technique turns these constrained optimizations of power control into convex optimization, which is intrinsically tractable despite its nonlinearity in objective and constraints. However, when SIR is comparable to or below 0 dB, the power control problems are truly nonconvex


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with no efficient and global solution methods. In this case, we present a heuristic that is provably convergent and empirically almost always computes the globally optimal power allocation by solving a sequence of GPs through the approach of successive convex approximations. The GP approach reveals the hidden convexity structure, which implies efficient solution methods and the global optimality of any local optimum in power control problems with nonlinear objective functions. It clearly differentiates the tractable formulations in a high-SIR regime from the intractable ones in a low-SIR regime. Power control by GP is applicable to formulations in both cellular networks with single-hop transmission between mobile users and base stations, and ad hoc networks with multihop transmission among the nodes, as illustrated through several numerical examples in this section. Traditionally, GP is solved by centralized computation through the highly efficient interior point methods. In this section we present a new result on how GP can be solved distributively with message passing, which has independent value to general maximization of coupled objective, and applies it to power control problems with a further reduction of message-passing overhead by leveraging the specific structures of power control problems. More generally, the technique of nonlinear change of variables, including the log change of variables, to reveal “hidden” convexity in optimization formulations has recently become quite popular in the communication network research community.

5.3.2 Geometric Programming GP is a class of nonlinear, nonconvex optimization problems with many useful theoretical and computational properties. It was invented in 1967 by Duffin, Peterson, and Zener [17], and much of the development by the early 1980s was summarized in [1]. Because a GP can be turned into a convex optimization problem, a local optimum is also a global optimum, the Lagrange duality gap is zero under mild conditions, and a global optimum can be computed very efficiently. Numerical efficiency holds both in theory and in practice: interior point methods applied to GP have provably polynomial-time complexity [48], and are very fast in practice with high-quality software downloadable from the Internet (e.g., the MOSEK package). Convexity and duality properties of GP are well understood, and large-scale, robust numerical solvers for GP are available. Furthermore, special structures in GP and its Lagrange dual problem lead to distributed algorithms, physical interpretations, and computational acceleration beyond the generic results for convex optimization. A detailed tutorial of GP and comprehensive survey of its recent applications to communication systems and to circuit design can be found in [11] and [3], respectively. This section contains a brief introduction of GP terminology.

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There are two equivalent forms of GP: standard form and convex form. The first is a constrained optimization of a type of function called posynomial, and the second form is obtained from the first through a logarithmic change of variable. We first define a monomial as a function f : Rn++ → R: (1)

(2)

f (x) = dxa1 xa2

(n)

. . . xan

,

where the multiplicative constant d ≥ 0 and the exponential constants a(j) ∈ R, j = 1, 2, . . . , n. A sum of monomials, indexed by k below, is called a posynomial: K (1) (2) (n) X a a a f (x) = dk x1 k x2 k . . . xnk , k=1

(j)

where dk ≥ 0, k = 1, 2, . . . , K, and ak ∈ R, j = 1, 2, . . . , n, k = 1, 2, . . . , K. 0.5 100 For example, 2x−π is a posynomial in x, x1 − x2 is not a 1 x2 + 3x1 x3 posynomial, and x1 /x2 is a monomial, thus also a posynomial. Minimizing a posynomial subject to posynomial upper bound inequality constraints and monomial equality constraints is called GP in standard form: minimize f0 (x) subject to fi (x) ≤ 1, i = 1, 2, . . . , m, hl (x) = 1, l = 1, 2, . . . , M, PKi

(5.13) (1)

(2)

aik aik k=1 dik x1 x2 (1) (2) (n) a a a dl x1 l x2 l . . . xnl .

where fi , i = 0, 1, . . . , m, are posynomials: fi (x) =

(n)

a

. . . xnik ,

and hl , l = 1, 2, . . . , M , are monomials: hl (x) = GP in standard form is not a convex optimization problem, because posynomials are not convex functions. However, with a logarithmic change of the variables and multiplicative constants: yi = log xi , bik = log dik , bl = log dl , and a logarithmic change of the functions’ values, we can turn it into the following equivalent problem in y. P 0 T minimize p0 (y) = log K k=1 exp(a0k y + b0k ) PK i subject to pi (y) = log k=1 exp(aTik y + bik ) ≤ 0, i = 1, 2, . . . , m, ql (y) = aTl y + bl = 0, l = 1, 2, . . . , M.

(5.14)

This is referred to as GP in convex form, which is a convex optimization problem because it can be verified that the log-sum-exp function is convex [4]. In summary, GP is a nonlinear, nonconvex optimization problem that can be transformed into a nonlinear convex problem. GP in standard form can be used to formulate network resource allocation problems with nonlinear objectives under nonlinear QoS constraints. The basic idea is that resources are often allocated proportional to some parameters, and when resource allo-


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5 120

4.5 4

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3.5 Function

Function

80 60

3 2.5 2

40

1.5 20 0 0 10

5 5

0 X

10

1 0.5 4

Y

4 2 B

2 0

0

A

Fig. 5.6 A bivariate posynomial before (left graph) and after (right graph) the log transformation. A nonconvex function is turned into a convex one.

cations are optimized over these parameters, we are maximizing an inverted posynomial subject to lower bounds on other inverted posynomials, which are equivalent to GP in standard form.

SP/GP, SOS/SDP Note that, although the posynomial seems to be a nonconvex function, it becomes a convex function after the log transformation, as shown in an example in Figure 5.6. Compared to the (constrained or unconstrained) minimization of a polynomial, the minimization of a posynomial in GP relaxes the integer constraint on the exponential constants but imposes a positivity constraint on the multiplicative constants and variables. There is a sharp contrast between these two problems: polynomial minimization is NP-hard, but GP can be turned into convex optimization with provably polynomial-time algorithms for a global optimum. In an extension of GP called signomial programming discussed later in this section, the restriction of nonnegative multiplicative constants is removed. This results in a general class of nonlinear and truly nonconvex problems that is simultaneously a generalization of GP and polynomial minimization over the positive quadrant, as summarized in the comparison Table 5.2.

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Table 5.2 Comparison of GP, constrained polynomial minimization over the positive quadrant (PMoP), and signomial programming (SP). All three types of problems minimize a sum of monomials subject to upper bound inequality constraints on sums of monomials, Q (j) but have different definitions of monomial: c j xa , as shown in the table. GP is known j to be polynomial-time solvable, but PMoP and SP are not. c a(j) xj

GP PMoP SP R+ R R R Z+ R R++ R++ R++

The objective function of signomial programming can be formulated as minimizing a ratio between two posynomials, which is not a posynomial (because posynomials are closed under positive multiplication and addition but not division). As shown in Figure 5.7, a ratio between two posynomials is a nonconvex function both before and after the log transformation. Although it does not seem likely that signomial programming can be turned into a convex optimization problem, there are heuristics to solve it through a sequence of GP relaxations. However, due to the absence of algebraic structures found in polynomials, such methods for signomial programming currently lack a theoretical foundation of convergence to global optimality. This is in contrast to the sum-of-squares method [51], which uses a nested family of SDP relax-

60

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40 Function

Function

3 20

2.5

0

−20 10

2 3 10 5 Y

2

5 0 0

X

3

2 1 B

1 0 0

A

Fig. 5.7 Ratio between two bivariate posynomials before (left graph) and after (right graph) the log transformation. It is a nonconvex function in both cases.


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ations to solve constrained polynomial minimization problems as explained in the last section.

5.3.3 Power Control by Geometric Programming: Convex Case Various schemes for power control, centralized or distributed, have been extensively studied since the 1990s based on different transmission models and application needs (e.g., in [2, 26, 47, 55, 63, 72]). This section summarizes the new approach of formulating power control problems through GP. The key advantage is that globally optimal power allocations can be efficiently computed for a variety of nonlinear systemwide objectives and user QoS constraints, even when these nonlinear problems appear to be nonconvex optimization.

Basic Model Consider a wireless (cellular or multihop) network with n logical transmitter/receiver pairs. Transmit powers are denoted as P1 , . . . , Pn . In the cellular uplink case, all logical receivers may reside in the same physical receiver, that is, the base station. In the multihop case, because the transmission environment can be different on the links comprising an end-to-end path, power control schemes must consider each link along a flow’s path. Under Rayleigh fading, the power received from transmitter j at receiver i is given by Gij Fij Pj where Gij ≥ 0 represents the path gain (it may also encompass antenna gain and coding gain) that is often modeled as proportional to d−γ ij , where dij denotes distance, γ is the power fall-off factor, and Fij model Rayleigh fading and are independent and exponentially distributed with unit mean. The distribution of the received power from transmitter j at receiver i is then exponential with mean value E [Gij Fij Pj ] = Gij Pj . The SIR for the receiver on logical link i is: SIRi = PN

j6=i

Pi Gii Fii Pj Gij Fij + ni

(5.15)

where ni is the noise power for receiver i. The constellation size M used by a link can be closely approximated for MQAM modulations as follows. M = 1+(−φ1 / (ln(φ2 BER))) SIR, where BER is the bit error rate and φ1 , φ2 are constants that depend on the modulation type. Defining K = −φ1 / (ln(φ2 BER)) leads to an expression of the data rate Ri on the ith link as a function of the SIR: Ri = (1/T ) log2 (1+KSIRi ), which can be approximated as

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Ri =

1 log2 (KSIRi ) T

(5.16)

when KSIR is much larger than 1. This approximation is reasonable either when the signal level is much higher than the interference level or, in CDMA systems, when the spreading gain is large. For notational simplicity in the rest of this section, we redefine Gii as K times the original Gii , thus absorbing constant K into the definition of SIR. The aggregate data rate for the system can then be written as " # Y X 1 Ri = log2 SIRi . Rsystem = T i i So in the high SIR regime, aggregate data rate maximization is equivalent to maximizing a product of SIR. The system throughput is the aggregate data rate supportable by the system given a set of users with specified QoS requirements. Outage probability is another important QoS parameter for reliable communication in wireless networks. A channel outage is declared and packets lost when the received SIR falls below a given threshold SIRth , often computed from the BER requirement. Most systems are interference-dominated and the thermal noise is relatively small, thus the ith link outage probability is Po,i = Prob{SIRi ≤ SIRth } = Prob{Gii Fii Pi ≤ SIRth

X j6=i

Gij Fij Pj }.

The outage probability can be expressed as [38] Po,i = 1 −

Y

j6=i 1 +

1 SIRth Gij Pj Gii Pi

,

which means that the upper bound Po,i ≤ Po,i,max can be written as an upper bound on a posynomial in P: ¶ Yµ SIRth Gij Pj 1 1+ ≤ . (5.17) Gii Pi 1 − Po,i,max j6=i

Cellular Wireless Networks We first present how GP-based power control applies to cellular wireless networks with one-hop transmission from N users to a base station. These results extend the scope of power control by the classical solution in CDMA


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systems that equalizes SIRs, and those by the iterative algorithms (e.g., in [2, 26, 47]) that minimize total power (a linear objective function) subject to SIR constraints. We start the discussion on the suite of power control problem formulations with a simple objective function and basic constraints. The following constrained problem of maximizing the SIR of a particular user i∗ is a GP. maximize Ri∗ (P) subject to Ri (P) ≥ Ri,min , ∀i, Pi1 Gi1 = Pi2 Gi2 , 0 ≤ Pi ≤ Pi,max , ∀i. The first constraint, equivalent to SIRi ≥ SIRi,min , sets a floor on the SIR of other users and protects these users from user i∗ increasing her transmit power excessively. The second constraint reflects the classical power control criterion in solving the near—far problem in CDMA systems: the expected received power from one transmitter i1 must equal that from another i2. The third constraint is regulatory or system limitations on transmit powers. All constraints can be verified to be inequality upper bounds on posynomials in transmit power vector P. Alternatively, we can use GP to maximize the minimum rate among all users. The maxmin fairness objective maximizeP min {Ri } i

can be accommodated in GP-based power control because it can be turned into equivalently maximizing an auxiliary variable t such that SIRi (P) ≥ exp(t), ∀i, which has a posynomial objective and constraints in (P, t). Example 5.5. A small illustrative example. A simple system comprised of five users is used for a numerical example. The five users are spaced at distances d of 1, 5, 10, 15, and 20 units from the base station. The power fall-off factor γ = 4. Each user has a maximum power constraint of Pmax = 0.5 mW. The noise power is 0.5 μW for all users. The SIR of all users, other than the user we are optimizing for, must be greater than a common threshold SIR level β. In different experiments, β is varied to observe the effect on the optimized user’s SIR. This is done independently for the near user at d = 1, a medium distance user at d = 15, and the far user at d = 20. The results are plotted in Figure 5.8. Several interesting effects are illustrated. First, when the required threshold SIR in the constraints is sufficiently high, there is no feasible power control solution. At moderate threshold SIR, as β is decreased, the optimized SIR initially increases rapidly. This is because it is allowed to increase its own power by the sum of the power reductions in the four other users, and the noise is relatively insignificant. At low threshold SIR, the noise becomes more significant and the power tradeoff from the other users less significant,

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Mung Chiang Optimized SIR vs. Threshold SIR 20 near medium far

15

Optimized SIR (dB)

10

5

0

−5

−10

−15

−20 −5

0

5

10

Threshold SIR (dB)

Fig. 5.8 Constrained optimization of power control in a cellular network (Example 5.5).

so the curve starts to bend over. Eventually, the optimized user reaches its upper bound on power and cannot utilize the excess power allowed by the lower threshold SIR for other users. This is exhibited by the transition from a sharp bend in the curve to a much shallower sloped curve. We now proceed to show that GP can also be applied to the problem formulations with an overall system objective of total system throughput, under both user data rate constraints and outage probability constraints. The following constrained problem of maximizing system throughput is a GP. maximize Rsystem (P) subject to Ri (P) ≥ Ri,min , ∀i, (5.18) Po,i (P) ≤ Po,i,max , ∀i, 0 ≤ Pi ≤ Pi,max , ∀i where the optimization variables are the transmit Q powers P. The objective is equivalent to minimizing the posynomial i ISRi , where ISR is 1/SIR. Each ISR is a posynomial in P and the product of posynomials is again a posynomial. The first constraint is from the data rate demand Ri,min by each user. The second constraint represents the outage probability upper bounds Po,i,max . These inequality constraints put upper bounds on posynomials of P, as can be readily verified through (5.16) and (5.17). Thus (5.18) is indeed a GP, and efficiently solvable for global optimality. There are several obvious variations of problem (5.18) that can be solved by GP; for example, we can lower bound Rsystem as Pa constraint and maximize Ri∗ for a particular user i∗ , or have a total power i Pi constraint or objective function.


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Table 5.3 Suite of power control optimization solvable by GP Objective Function (A) Max Ri∗ (specific user) (B) Max mini Ri (worst-case user) P (C) Max i Ri (total throughput) P (D) Max i wi Ri (weighted rate sum) P (E) Min i Pi (total power)

Constraints (a) Ri ≥ Ri,min (rate constraint) (b) Pi1 Gi1 = Pi2 Gi2 (near—far constraint) P (c) i Ri ≥ Rsystem,min (sum rate constraint) (d) Po,i ≤ Po,i,max (outage probability constraint) (e) 0 ≤ Pi ≤ Pi,max (power constraint)

The objective function toPbe maximized can also be generalized to a weighted sum of data rates, i wi Ri , where P w º 0 is a given weight vector. This is stillQa GP because maximizing i wi log SIRi is equivalent to maxiQ wi i mizing log i SIRw to minimizing ISR i , which is in turn equivalent i . Now i Q wi use auxiliary variables {ti }, and minimize i ti over the original constraints in (5.18) plus the additional constraints ISRi ≤ ti for all i. This is readily verified to be a GP in (x, t), and is equivalent to the original problem. Generalizing the above discussions and observing that high-SIR assumption is needed for GP formulation only when there are sums of log(1 + SIR) in the optimization problem, we have the following summary. Proposition 5.1. In the high-SIR regime, any combination of objectives (A)—(E) and constraints (a)—(e) in Table 5.3 (pick any one of the objectives and any subset of the constraints) is a power control optimization problem that can be solved by GP, that is, can be transformed into a convex optimization with efficient algorithms to compute the globally optimal power vector. When objectives (C)—(D) or constraints (c)—(d) do not appear, the power control optimization problem can be solved by GP in any SIR regime. In addition to efficient computation of the globally optimal power allocation with nonlinear objectives and constraints, GP can also be used for admission control based on feasibility study described in [11], and for determining which QoS constraint is a performance bottleneck, that is, met tightly at the optimal power allocation.3

3

This is because most GP solution algorithms solve both the primal GP and its Lagrange dual problem, and by the complementary slackness condition, a resource constraint is tight at optimal power allocation when the corresponding optimal dual variable is nonzero.

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Extensions In wireless multihop networks, system throughput may be measured either by end-to-end transport layer utilities or by link layer aggregate throughput. GP application to the first approach has appeared in [10], and those to the second approach in [11]. Furthermore, delay and buffer overflow properties can also be accommodated in the constraints or objective function of GP-based power control.

5.3.4 Power Control by Geometric Programming: Nonconvex Case If we maximize the total throughput Rsystem in the medium to low SIR case (i.e., when SIR is not much larger than 0 dB), the approximation of log(1 + SIR) as log SIR does not hold. Unlike SIR, which is an inverted posynomial, 1+ SIR is not an inverted posynomial. Instead, 1/ (1 + SIR) is a ratio between two posynomials: P f (P) j6=i Gij Pj + ni . (5.19) = P g(P) j Gij Pj + ni

Minimizing, or upper bounding, a ratio between two posynomials belongs to a truly nonconvex class of problems known as complementary GP [1, 11] that is in general an NP-hard problem. An equivalent generalization of GP is signomial programming [1, 11]: minimizing a signomial subject to upper bound inequality constraints on signomials, where a signomial s(x) is a sum coefficients: P of monomials, possiblyN with negative multiplicative s(x) = N and gi (x) are monomials.4 i=1 ci gi (x) where c ∈ R Successive Convex Approximation Method Consider the following nonconvex problem, minimize f0 (x) subject to fi (x) ≤ 1, i = 1, 2, . . . , m,

(5.20)

where f0 is convex without loss of generality,5 but the fi (x)s, ∀i are nonconvex. Because directly solving this problem is NP-hard, we want to solve it by 4

An SP can always be converted into a complementary GP, because an inequality in SP, which can be written as fi1 (x) − fi2 (x) ≤ 1, where fi1 , fi2 are posynomials, is equivalent to an inequality fi1 (x)/ (1 + fi2 (x)) ≤ 1 in complementary GP. 5 If f is nonconvex, we can move the objective function to the constraint by introducing 0 auxiliary scalar variable t and writing minimize t subject to the additional constraint f0 (x) − t ≤ 0.


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a series of approximations f˜i (x) ≈ fi (x), ∀x, each of which can be optimally solved in an easy way. It is known [46] that if the approximations satisfy the following three properties, then the solutions of this series of approximations converge to a point satisfying the necessary optimality Karush—Kuhn—Tucker (KKT) conditions of the original problem. (1) fi (x) ≤ f˜i (x) for all x. (2) fi (x0 ) = f˜i (x0 ) where x0 is the optimal solution of the approximated problem in the previous iteration. (3) ∇fi (x0 ) = ∇f˜i (x0 ). The following algorithm describes the generic successive approximation approach. Given a method to approximate fi (x) with f˜i (x) , ∀i, around some point of interest x0 , the following algorithm provides the output of a vector that satisfies the KKT conditions of the original problem. Algorithm 2. Successive approximation to a nonconvex problem. 1. Choose an initial feasible point x(0) and set k = 1. 2. Form an approximated problem of (5.20) based on the previous point x(k−1) . 3. Solve the kth approximated problem to obtain x(k) . 4. Increment k and go to step 2 until convergence to a stationary point. Single condensation method. Complementary GPs involve upper bounds on the ratio of posynomials as in (5.19); they can be turned into GPs by approximating the denominator of the ratio of posynomials, g(x), with a monomial g˜(x), but leaving the numerator f (x) as a posynomial. The following basic result can be readily proved using the arithmeticmean—geometric-mean inequality. P Lemma 5.1. Let g(x) = i ui (x) be a posynomial. Then Y µ ui (x) ¶αi g(x) ≥ g˜(x) = . (5.21) αi i If, in addition, αi = ui (x0 )/g(x0 ), ∀i, for any fixed positive x0 , then g˜(x0 ) = g(x0 ), and g˜(x) is the best local monomial approximation to g(x) near x0 in the sense of first-order Taylor approximation. The above lemma easily leads to the following Proposition 5.2. The approximation of a ratio of posynomials f (x)/g(x) with f (x)/˜ g (x) where g˜(x) is the monomial approximation of g(x) using the arithmetic-geometric mean approximation of Lemma 5.1 satisfies the three conditions for the convergence of the successive approximation method. Double condensation method. Another choice of approximation is to make a double monomial approximation for both the denominator and numerator in (5.19). However, in order to satisfy the three conditions for the convergence

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of the successive approximation method, a monomial approximation for the numerator f (x) should satisfy f (x) ≤ f˜(x). Applications to Power Control Figure 5.9 shows a block diagram of the approach of GP-based power control for a general SIR regime [64]. In the high SIR regime, we need to solve only one GP. In the medium to low SIR regimes, we solve truly nonconvex power control problems that cannot be turned into convex formulation through a series of GPs.

(High SIR)

Original Problem

- Solve

(Medium to Low SIR)

Original Problem

- SP - Complementary - Solve 1 GP GP (Condensed)

1 GP

6

Fig. 5.9 GP-based power control in different SIR regimes.

GP-based power control problems in the medium to low SIR regimes become SP (or, equivalently, complementary GP), which can be solved by the single or double condensation method. We focus on the single condensation method here. Consider a representative problem formulation of maximizing total system throughput in a cellular wireless network subject to user rate and outage probability constraints in problem (5.18), which can be explicitly written out as QN 1 minimize i=1 1+SIRi 1 subject to (2T Ri,min − 1) SIR ≤ 1, i = 1, . . . , N, i (5.22) Q Gij Pj N −1 (1 − Po,i,max ) N (SIRth ) j6=i Gii Pi ≤ 1, i = 1, . . . , N, Pi (Pi,max )−1 ≤ 1, i = 1, . . . , N. All the constraints are posynomials. However, the objective is not a posynomial, but a ratio between two posynomials as in (5.19). This power control problem can be solved by the condensation method by solving a series of GPs. Specifically, we have the following single-condensation algorithm. Algorithm 3. Single condensation GP power control. 1. Evaluate the denominator posynomial of the objective function in (5.22) with the given P.


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Optimized total system throughput

Total system throughput achieved

5300

5250

5200

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5100

5050 0

50

100

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200 250 300 Experiment index

350

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Fig. 5.10 Maximized total system throughput achieved by the (single) condensation method for 500 different initial feasible vectors (Example 5.6). Each point represents a different experiment with a different initial power vector.

2. Compute for each term i in this posynomial, αi =

value of ith term in posynomial . value of posynomial

3. Condense the denominator posynomial of the (5.22) objective function into a monomial using (5.21) with weights αi . 4. Solve the resulting GP using an interior point method. 5. Go to step 1 using P of step°4. ° 6. Terminate the kth loop if °P(k) − P(k−1) ° ≤ where is the error tolerance for exit condition. As condensing the objective in the above problem gives us an underestimate of the objective value, each GP in the condensation iteration loop tries to improve the accuracy of the approximation to a particular minimum in the original feasible region. All three conditions for convergence are satisfied, and the algorithm is convergent. Empirically through extensive numerical experiments, we observe that it almost always computes the globally optimal power allocation.

Example 5.6. Single condensation example. We consider a wireless cellular network with three users. Let T = 10−6 s, Gii = 1.5, and generate Gij , i 6= j, as independent random variables uniformly distributed between 0 and 0.3. Threshold SIR is SIRth = −10 dB, and minimal data rate requirements are 100 kbps, 600 kbps, and 1000 kbps for logical links 1, 2, and 3, respectively.

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Maximal outage probabilities are 0.01 for all links, and maximal transmit powers are 3 mW, 4 mW, and 5 mW for links 1, 2, and 3, respectively. For each instance of problem (5.22), we pick a random initial feasible power vector P uniformly between 0 and Pmax . Figure 5.10 compares the maximized total network throughput achieved over 500 sets of experiments with different initial vectors. With the (single) condensation method, SP converges to different optima over the entire set of experiments, achieving (or coming very close to) the global optimum at 5290 bps 96% of the time and a local optimum at 5060 bps 4% of the time. The average number of GP iterations required by the condensation method over the same set of experiments is 15 if an extremely tight exit condition is picked for SP condensation iteration: = 1 × 10−10 . This average can be substantially reduced by using a larger ; for example, increasing to 1 × 10−2 requires on average only 4 GPs. We have thus far discussed a power control problem (5.22) where the objective function needs to be condensed. The method is also applicable if some constraint functions are signomials and need to be condensed [14, 35].

5.3.5 Distributed Algorithm A limitation for GP-based power control in ad hoc networks (without base stations) is the need for centralized computation (e.g., by interior point methods). The GP formulations of power control problems can also be solved by a new method of distributed algorithm for GP. The basic idea is that each user solves its own local optimization problem and the coupling among users is taken care of by message passing among the users. Interestingly, the special structure of coupling for the problem at hand (all coupling among the logical links can be lumped together using interference terms) allows one to further reduce the amount of message passing among the users. Specifically, we use a dual decomposition method to decompose a GP into smaller subproblems whose solutions are jointly and iteratively coordinated by the use of dual variables. The key step is to introduce auxiliary variables and to add extra equality constraints, thus transferring the coupling in the objective to coupling in the constraints, which can be solved by introducing “consistency pricing” (in contrast to “congestion pricing”). We illustrate this idea through an unconstrained GP followed by an application of the technique to power control.

Distributed Algorithm for GP Suppose we have the following unconstrained standard form GP in x Â 0, P minimize i fi (xi , {xj }j∈I(i) ), (5.23)


169

where xi denotes the local variable of the ith user, {xj }j∈I(i) denotes the coupled variables from other users, and fi is either a monomial or posynomial. Making a change of variable yi = log xi , ∀i, in the original problem, we obtain P minimize i fi (eyi , {eyj }j∈I(i) ).

We now rewrite the problem by introducing auxiliary variables yij for the coupled arguments and additional equality constraints to enforce consistency: P yi yij }j∈I(i) ) minimize i fi (e , {e (5.24) subject to yij = yj , ∀j ∈ I(i), ∀i.

Each ith user controls the local variables (yi , {yij }j∈I(i) ). Next, the Lagrangian of (5.24) is formed as X X X L({yi }, {yij }; {γij }) = fi (eyi , {eyij }j∈I(i) ) + γij (yj − yij ) i

=

X i

i

j∈I(i)

Li (yi , {yij }; {γij }),

where yi

yij

Li (yi , {yij }; {γij }) = fi (e , {e

}j∈I(i) ) +

µ X

j:i∈I(j)

¶

γji yi −

X

γij yij .

j∈I(i)

(5.25) The minimization of the Lagrangian with respect to the primal variables ({yi }, {yij }) can be done simultaneously and distributively by each user in parallel. In the more general case where the original problem (5.23) is constrained, the additional constraints can be included in the minimization at each Li . In addition, the following master Lagrange dual problem has to be solved to obtain the optimal dual variables or consistency prices {γij }, max g({γij }),

{γij }

where g({γij }) =

X i

(5.26)

min Li (yi , {yij }; {γij }).

yi ,{yij }

Note that the transformed primal problem (5.24) is convex with zero duality gap; hence the Lagrange dual problem indeed solves the original standard GP problem. A simple way to solve the maximization in (5.26) is with the following subgradient update for the consistency prices, γij (t + 1) = γij (t) + δ(t)(yj (t) − yij (t)).

(5.27)

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Appropriate choice of the stepsize δ(t) > 0, for example, δ(t) = δ0 /t for some constant δ0 > 0, leads to convergence of the dual algorithm. Summarizing, the ith user has to: (i) minimize the function Li in (5.25) involving only local variables, upon receiving the updated dual variables {γji , j : i ∈ I(j)}, and (ii) update the local consistency prices {γij , j ∈ I(i)} with (5.27), and broadcast the updated prices to the coupled users.

Applications to Power Control As an illustrative example, we maximize the total system throughput in the high SIR regime with constraints local to each user. If we directly applied the distributed approach described in the last section, the resulting algorithm would require knowledge by each user of the interfering channels and interfering transmit powers, which would translate into a large amount of message passing. To obtain a practical distributed solution, we can leverage the structures of power control problems at hand, and instead keep a local copy of each of the effective received powers PijR = Gij Pj . Again using problem (5.18) as an example formulation and assuming high SIR, we can write the problem as (after the log change of variable) ³ ³ ´´ P −1 ˜ P exp(P˜ R ) + σ 2 minimize ij i log Gii exp(−Pi ) j6=i (5.28) ˜ ij + P˜j , subject to P˜ R = G ij

Constraints are local to each user, for example, (a), (d), and (e) in Table 5.3. The partial Lagrangian is ⎛ ⎛ ⎞⎞ X X ˜ ⎝ log⎝G−1 exp(P˜ijR ) + σ2 ⎠⎠ L= ii exp(−Pi ) i

+

j6=i

XX i

j6=i

³ ³ ´´ ˜ ij + P˜j , γij P˜ijR − G

(5.29)

and the local ith Lagrangian function in (5.29) is distributed to the ith user, from which the dual decomposition method can be used to determine the optimal power allocation P∗ . The distributed power control algorithm is summarized as follows. Algorithm 4. Distributed power allocation update to maximize Rsystem . At each iteration t: ³P ´ 1. The ith user receives the term γ (t) involving the dual varij6=i ji ables from the interfering users by message-passing and n o minimizes the following local Lagrangian with respect to P˜i (t), P˜ijR (t) subject to the local j

constraints.


µ

171

¶

n o Li P˜i (t), P˜ijR (t) ; {γij (t)}j j ⎛ ⎛ ⎞⎞ X ˜ ⎝ = log⎝G−1 exp(P˜ijR (t)) + σ 2 ⎠⎠ ii exp(−Pi (t)) j6=i

⎛ ⎞ X X + γij P˜ijR (t) − ⎝ γji (t)⎠ P˜i (t). j6=i

j6=i

2. The ith user estimates the effective received power from each of the interfering users PijR (t) = Gij Pj (t) for j 6= i, updates the dual variable by ³ ´ γij (t + 1) = γij (t) + (δ0 /t) P˜ijR (t) − log Gij Pj (t) ,

(5.30)

and then broadcasts them by message passing to all interfering users in the system. Example 5.7. Distributed GP power control. We apply the distributed algorithm to solve the above power control problem for three logical links with Gij = 0.2, i 6= j, Gii = 1, ∀i, maximal transmit powers of 6 mW, 7 mW, and 7 mW for links 1, 2, and 3 respectively. Figure 5.11 shows the convergence of the dual objective function towards the globally optimal total throughput of the network. Figure 5.12 shows the convergence of the two auxiliary variables in links 1 and 3 towards the optimal solutions.

4

2.2

Dual objective function

x 10

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

0

50

100 Iteration

150

200

Fig. 5.11 Convergence of the dual objective function through distributed algorithm (Example 5.7).

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Mung Chiang Consistency of the auxiliary variables 2

0

−2

−4

log(P )

−6

2 R

log(P12/G ) 12

R

−8

−10

log(P32/G32)

0

50

100 Iteration

150

200

Fig. 5.12 Convergence of the consistency constraints through distributed algorithm (Example 5.7).

5.3.6 Concluding Remarks and Future Directions Power control problems with nonlinear objective and constraints may seem to be difficult, NP-hard problems to solve for global optimality. However, when SIR is much larger than 0 dB, GP can be used to turn these problems into intrinsically tractable convex formulations, accommodating a variety of possible combinations of objective and constraint functions involving data rate, delay, and outage probability. Then interior point algorithms can efficiently compute the globally optimal power allocation even for a large network. Feasibility analysis of GP naturally leads to admission control and pricing schemes. When the high SIR approximation cannot be made, these power control problems become SP and may be solved by the heuristic of the condensation method through a series of GPs. Distributed optimal algorithms for GP-based power control in multihop networks can also be carried out through message passing. Several challenging research issues in the low-SIR regime remain to be further explored. These include, for example, the reduction of SP solution complexity (e.g., by using high-SIR approximation to obtain the initial power vector and by solving the series of GPs only approximately except the last GP), and the combination of SP solution and distributed algorithm for distributed power control in low SIR regime. We also note that other approaches to tackle nonconvex power control problems have been studied, for example, the use of a particular utility function of rate to turn the problem into a convex one [28].


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5.4 DSL Spectrum Management 5.4.1 Introduction Digital subscriber line (DSL) technologies transform traditional voice-band copper channels into high-bandwidth data pipes, which are currently capable of delivering data rates up to several Mbps per twisted-pair over a distance of about 10 kft. The major obstacle for performance improvement in today’s DSL systems (e.g., ADSL and VDSL) is crosstalk, which is the interference generated between different lines in the same binder. The crosstalk is typically 10—20 dB larger than the background noise, and direct crosstalk cancellation (e.g., [6, 27]) may not be feasible in many cases due to complexity issues or as a result of unbundling. To mitigate the detriments caused by crosstalk, static spectrum management which mandates spectrum mask or flat power backoff across all frequencies (i.e., tones) has been implemented in the current system. Dynamic spectrum management (DSM) techniques, on the other hand, can significantly improve data rates over the current practice of static spectrum management. Within the current capability of the DSL modems, each modem has the capability to shape its own power spectrum density (PSD) across different tones, but can only treat crosstalk as background noise (i.e., no signal level coordination, such as vector transmission or iterative decoding, is allowed), and each modem is inherently a single-input—single-output communication system. The objective would be to optimize the PSD of all users on all tones (i.e., continuous power loading or discrete bit loading), such that they are “compatible” with each other and the system performance (e.g., weighted rate sum as discussed below) is maximized. Compared to power control in wireless networks treated in the last section, the channel gains are not time-varying in DSL systems, but the problem dimension increases tremendously because there are many “tones” (or frequency carriers) over which transmission takes place. Nonconvexity still remains a major technical challenge, and high SIR approximation in general cannot be made. However, utilizing the specific structures of the problem (e.g., the interference channel gain values), an efficient and distributed heuristic is shown to perform close to the optimum in many realistic DSL network scenarios. Following [7], this section presents a new algorithm for spectrum management in frequency selective interference channels for DSL, called autonomous spectrum balancing (ASB). It is the first DSL spectrum management algorithm that satisfies all of the following requirements for performance and complexity. It is autonomous (distributed algorithm across the users without explicit information exchange) with linear-complexity, while provably convergent, and comes close to the globally optimal rate region in practice. ASB

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Mung Chiang

overcomes the bottlenecks in the state-of-the-art algorithms in DSM, including IW, OSB, and ISB summarized below. Let K be the number of tones and N the number of users (lines). The iterative waterfilling (IW) algorithm [74] is among one of the first DSM algorithms proposed. In IW, each user views any crosstalk experienced as additive Gaussian noise, and seeks to maximize its data rate by “waterfilling” over the aggregated noise plus interference. No information exchange is needed among users, and all the actions are completely autonomous. IW leads to a great performance increase over the static approach, and enjoys a low complexity that is linear in N . However, the greedy nature of IW leads to a performance far from optimal in the near—far scenarios such as mixed CO/RT deployment and upstream VDSL. To address this, an optimal spectrum balancing (OSB) algorithm [9] has been proposed, which finds the best possible spectrum management solution under the current capabilities of the DSL modems. OSB avoids the selfish behaviors of individual users by aiming at the maximization of a total weighted sum of user rates, which corresponds to a boundary point of the achievable rate region. On the other hand, OSB has a high computational complexity that is exponential in N , which quickly leads to intractability when N is larger than 6. Moreover, it is a completely centralized algorithm where a spectrum management center at the central office needs to know the global information (i.e., all the noise PSDs and crosstalk channel gains in the same binder) to perform the algorithm. As an improvement to the OSB algorithm, an iterative spectrum balancing (ISB) algorithm [8] has been proposed, which is based on a weighted sum rate maximization similar to OSB. Different from OSB, ISB performs the optimization iteratively through users, which leads to a quadratic complexity in N . Close to optimal performance can be achieved by the ISB algorithm in most cases. However, each user still needs to know the global information as in OSB, thus ISB is still a centralized algorithm and is considered to be impractical in many cases. This section presents the ASB algorithm [7], which attains near-optimal performance in an implementable way. The basic idea is to use the concept of a reference line to mimic a “typical” victim line in the current binder. By setting the power spectrum level to protect the reference line, a good balance between selfish and global maximizations can be achieved. The ASB algorithm enjoys a linear complexity in N and K, and can be implemented in a completely autonomous way. We prove the convergence of ASB under both sequential and parallel updates. Table 5.4 compares various aspects of different DSM algorithms. Utilizing the structures of the DSL problem, in particular, the lack of channel variation and user mobility, is the key to provide a linear complexity, distributed, convergent, and almost optimal solution to this coupled, nonconvex optimization problem.


175

Table 5.4 Comparison of different DSM algorithms Algorithm Operation

Complexity Performance Reference

IW OSB ISB ASB

O (KN) ¢ ¡ O KeN ¢ ¡ O KN 2 O (KN)

Autonomous Centralized Centralized Autonomous

Suboptimal Optimal Near optimal Near optimal

[74] [9] [8] [7]

5.4.2 System Model Using the notation as in [9, 8], we consider a DSL bundle with N = {1, . . . , N } modems (i.e., lines, users) and K = {1, . . . , K} tones. Assume discrete multitone (DMT) modulation is employed by all modems; transmission can be modeled independently on each tone as yk = Hk xk + zk . The vector xk = {xnk , n ∈ N } contains transmitted signals on tone k, where xnk is the signal transmitted onto line n at tone k. yk and zk have similar structures. yk is the vector of received signals on tone k. zk is the vector of additive noise on tone k and contains thermal noise, alien crosstalk, singlecarrier modems, radio frequency interference, and so on. Hk = [hn,m ]n,m∈N is k is the channel from the N × N channel transfer matrix on tone k, where hn,m k TX m to RX n on tone k. The diagonal elements of Hk contain the directchannels whereas the off-diagonal elements contain the crosstalk We n channels. o n n 2 denote the transmit power spectrum density (PSD) sk = E |xk | . In the last section’s notation for single-carrier systems, we would have snk = Pn , ∀k. For convenience we denote the vector containing the PSD of user n on all tones as sn = {snk , k ∈ K}. We denote the DMT symbol rate as fs . Assume that each modem treats interference from other modems as noise. When the number of interfering modems is large, the interference can be well approximated by a Gaussian distribution. Under this assumption the achievable bit loading of user n on tone k is Ã ! 1 snk n bk = log 1 + P , (5.31) n Γ m6=n αn,m sm k + σk k 2

2

= |hn,m | / |hn,n where αn,m k k k | is the normalized crosstalk channel gain, and 2 n σk is the noise power density normalized by the direct channel gain |hn,n k | . Here Γ denotes the SINR-gap to capacity, which is a function of the desired BER, coding gain, and noise margin [61]. Without loss of generality, we assume Γ = 1. The data rate on line n is thus

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Mung Chiang

Rn = fs

X

bnk .

(5.32)

k∈K

Each modem n is typically subject to a total power constraint P n , due to the limitations on each modem’s analogue frontend. X snk ≤ P n . (5.33) k∈K

5.4.3 Spectrum Management Problem Formulation One way to define the spectrum management problem is start with the following optimization problem. maximize R1 n n,target subject to R , ∀n > 1 P ≥ Rn n k∈K sk ≤ P , ∀n.

(5.34)

Here Rn,target is the target rate constraint of user n. In other words, we try to maximize the achievable rate of user 1, under the condition that all other users achieve their target rates Rn,target . The mutual interference in (5.31) causes Problem (5.34) to be coupled across users on each tone, and the individual total power constraint causes Problem (5.34) to be coupled across tones as well. Moreover, the objective function in Problem (5.34) is nonconvex due to the coupling of interference, and the convexity of the rate region cannot be guaranteed in general. However, it has been shown in [75] that the duality gap between the dual gap of Problem (5.34) goes to zero when the number of tones K gets large (e.g., for VDSL), thus Problem (5.34) can be solved by the dual decomposition method, which brings the complexity as a function of K down to linear. Moreover, a frequency-sharing property ensures the rate region is convex with large enough K, and each boundary point of the boundary point of the rate region can be achieved by a weighted rate maximization as (following [9]), P 1 n n maximize R P + nn>1 w nR (5.35) subject to k∈K sk ≤ P , ∀n ∈ N , such that the nonnegative weight coefficient wn is adjusted to ensure that the target rate constraint of user n is met. Without loss of generality, here we define w1 = 1. By changing the rate constraints Rn,target for users n > 1 (or equivalently, changing the weight coefficients, wn for n > 1), every boundary point of the convex rate region can be traced.


177

We observe that at the optimal solutions of (5.34), each user chooses a PSD level that leads to a good balance of maximization of its own rate and minimization of the damages it causes to the other users. To accurately calculate the latter, the user needs to know the global information of the noise PSDs and crosstalk channel gains. However, if we aim at a less aggressive objective and only require each user give enough protection to the other users in the binder while maximizing her own rate, then global information may not be needed. Indeed, we can introduce the concept of a “reference line”, a virtual line that represents a “typical” victim in the current binder. Then instead of solving (5.34), each user tries to maximize the achievable data rate on the reference line, subject to its own data rate and total power constraint. Define the rate of the reference line to user n as ¶ µ X X s˜k n,ref n ˜ R . = log 1 + n n bk = α ˜ k sk + σ ˜k k∈K

k∈K

The coefficients {˜ sk , σ ˜k , α ˜ nk , ∀k, n} are parameters of the reference line and can be obtained from field measurement. They represent the conditions of a “typical” victim user in an interference channel (here a binder of DSL lines), and are known to the users a priori. They can be further updated on a much slower time scale through channel measurement data. User n then wants to solve the following problem local to itself, maximize Rn,ref n n,target subject to R , P ≥ Rn n s ≤ P . k∈K k

(5.36)

By using Lagrangian relaxation on the rate target constraint in Problem (5.36) with a weight coefficient (dual variable) wn , the relaxed version of (5.36) is n n n,ref maximize w P R +nR n (5.37) subject to k∈K sk ≤ P . The weight coefficient wn needs to be adjusted to enforce the rate constraint.

5.4.4 ASB Algorithms We first introduce the basic version of the ASB algorithm (ASB-I), where each user n chooses the PSD sn to solve (5.36) , and updates the weight coefficient wn to enforce the target rate constraint. Then we introduce a variation of the ASB algorithm (ASB-II) that enjoys even lower computational complexity and provable convergence.

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Mung Chiang

ASB-I For each user n, replacing the original optimization (5.36) with the Lagrange dual problem X ¡ ¢ max n n max Jkn wn , λn , snk , s−n , (5.38) P k n n λ ≥0,

where

k∈K

sk ≤P

k∈K

sk

¡ ¢ Jkn wn , λn , snk , s−n = wn bnk + ˜bnk − λn snk . k

(5.39)

By introducing the dual variable λn , we decouple (5.36) into several smaller subproblems, one for each tone. And define Jkn as user n’s objective function on tone k. The optimal PSD that maximizes Jkn for given wn and λn is ¡ n n −n ¢ sn,I w , λ , sk = arg k

max

n sn k ∈[0,P ]

¡ ¢ Jkn wn , λn , snk , s−n , k

(5.40)

which can be found by solving the first-order condition, ¡ ¢ ∂Jkn wn , λn , snk , s−n /∂snk = 0, k

which leads to

α ˜ nk s˜k ³ ´³ ´ − λn = 0. − n,m m n n sn,I + σ n sn,I + σ sn,I + α s + σ + α ˜ ˜ ˜ s ˜ α ˜ m6=n k k k k k k k k k k k (5.41) Note that (5.41) can be simplified into a cubic equation that has three solutions. The optimal PSD can be found by substituting these three solutions ¡ ¢ back to the objective function Jkn wn , λn , snk , s−n , as well as checking the k boundary solutions snk = 0 and snk = P n , and picking the one that yields the largest value of Jkn . The user then updates λn to enforce the power constraint, and updates n w to enforce the target rate constraint. The complete algorithm is given as follows, where ελ and εw are small stepsizes for updating λn and wn . P

wn

Algorithm 5. Autonomous Spectrum Balancing. repeat for each user n = 1, . . . , N repeat for each tone k = 1, . . . , K, find sn,I = arg maxsnk ≥0 Jkn hk ³P í+ n,I n λn = λn + ελ ; k sk − P P n + n n n,target − k bk )] ; w = [w + εw (R until convergence end until convergence


179

ASB-II with Frequency-Selective Waterfilling To obtain the optimal PSD in ASB-I (for fixed λn and wn ), we have to solve the roots of a cubic equation. To reduce the computational complexity and gain more insights of the solution structure, we assume that the reference line operates in the high SIR regime whenever it is active: If s˜k > 0, then s˜k À σ ˜k À αn,m snk for any feasible snk , n ∈ N , and k ∈ K. This assumption k is motivated by our observations on optimal solutions in the DSL type of interference channels. It means that the reference PSD is much larger than the reference noise, which is in turn much larger than the interference from ¯ = {k | s˜k > 0, k ∈ K}, the reference line’s user n. Then on any tone k ∈ K achievable rate is µ ¶ µ ¶ s˜k α ˜ n sn s˜k log 1 + n n ≈ log − k k, α ˜ k sk + σ ˜k σ ˜k σ ˜k and user n’s objective function on tone k can be approximated by µ ¶ ¡ ¢ α ˜ nk snk s˜k n n n n b − − λ s + log . = w Jkn,II,1 wn , λn , snk , s−n k k k σ ˜k σ ˜k

The corresponding optimal PSD is ⎛ ¡ ¢ sn,II,1 wn , λn , s−n =⎝ k k

⎞+ X n,m wn n⎠ − αk sm . k − σk λn + α ˜ kn /˜ σk

(5.42)

m6=n

This is a waterfilling type of solution and is intuitively satisfying: the PSD should be smaller when the power constraint is tighter (i.e., λn is larger), or the interference coefficient to the reference line α ˜ kn is higher, or the noise level on the reference line σ ˜ is smaller, or there is more interference plus noise k P n,m m n α s + σ on the current tone. It is different from the conventional k k m6=n k waterfilling in that the water level in each tone is not only determined by the dual variables wn and λn , but also by the parameters of the reference line, α ˜ kn /˜ σk . On the other hand, on any tone where the reference line is inactive, that ¯ C = {k | s˜k = 0, k ∈ K}, the objective function is is, k ∈ K ¡ ¢ Jkn,II,2 wn , λn , snk , s−n = wn bnk − λn snk , k and the corresponding optimal PSD is

⎞+ ⎛ n X ¡ ¢ w n⎠ αkn,m sm . sn,II,2 wn , λn , s−n =⎝ n − k − σk k k λ m6=n

This is the same solution as the iterative waterfilling.

(5.43)

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Mung Chiang

The choice of optimal PSD in ASB-II can be summarized as the following. ⎧³ ´+ P n,m m n ¯ ⎨ n wnn − α s − σ , k ∈ K, ¡ ¢ k k m6=n k λ +α ˜ k /˜ σk n n −n ´ ³ , λ , s w = sn,II + k k n,m m ⎩ wn − P ¯C. sk − σkn , k∈K m6=n αk λn (5.44) This is essentially a waterfilling type of solution, with different water levels for different tones (frequencies). We call it frequency selective waterfilling.

5.4.5 Convergence Analysis In this section, we show the convergence for both ASB-I and ASB-II, for the case where users fix their weight coefficients wn , which is also called rate adaptive (RA) spectrum balancing [61] that aims at maximizing users’ rates subject to power constraint.

Convergence in the Two-User Case first ¡The ¢ result is ¡on the¢ convergence of ASB-I algorithm, with fixed w = w1 , w2 and λ = λ1 , λ2 .

Proposition 5.3. The ASB-I algorithm converges in a¡ two-user ¢ case ¡ under ¢ 1 2 fixed w and λ, if users start from initial PSD values s , s = 0, P 2 or k k ¡ 1 2¢ ¡ 1 ¢ sk , sk = P , 0 on all tones.

The proof of Proposition 5.3 uses supermodular game theory [65] and strategy transformation similar to [32]. Now consider the ASB-II algorithm where two users sequentially optimize their PSD levels under fixed values of w, but adjust λ to enforce the power constraint. Denote sn,t k as the PSD of user n in tone k after iteration t, where P n,t n s = P is satisfied for any n and t. One iteration is defined as one k k round of updates of all users. We can show that Proposition 5.4. The ASB-II algorithm globally converges to the unique fixed point in a two-user system under fixed w, if maxk αk2,1 maxk α1,2 k < 1.

The convergence result of iterative waterfilling in the two-user case [74] is a special case of Proposition 5.4 by setting s˜k = 0, ∀k. We further extend the convergence results to a system with an arbitrary N > 2 of users. We consider both sequential and parallel PSD updates of the users. In the more realistic but harder-to-analyze parallel updates, time is divided into slots, and each user n updates the PSD simultaneously in each time slot according to (5.44) based on the PSDs in the previous slot, where the λn is adjusted such that the power constraint is satisfied.


User 1 User 2 User 3 User 4

CO 2Km

5Km

3Km 4Km

CP 4Km

RT RT

181

CP

3.5Km

RT

3Km

CP CP

Fig. 5.13 An example of mixed CO/RT deployment topology (Example 5.8).

Proposition 5.5. Assume maxn,m,k αkn,m < 1/ (N − 1); then the ASB-II algorithm globally converges (to the unique fixed point) in an N-user system under fixed w, with either sequential or parallel updates. Proposition 5.5 contains the convergence of iterative waterfilling in an N user case with sequential updates (proved in [15]) as a special case of ASB convergence with sequential or parallel updates. Moreover, the convergence proof for the parallel updates turns out to be simpler than the one for sequential updates. The proof extends that of Proposition 5.4, and can be found in [7].

5.4.6 Simulation Results Example 5.8. Mixed CO-RT DSL. Here we summarize a typical numerical example comparing the performances of ASB algorithms with IW, OSB, and ISB. We consider a standard mixed central office (CO) and remote terminal (RT) deployment. A four-user scenario has been selected to make a comparison with the highly complex OSB algorithm possible. As depicted in Figure 5.13 the scenario consists of one CO distributed line, and three RT distributed lines. The target rates on RT1 and RT2 have both been set to 2 Mbps. For a variety of different target rates on RT3, the CO line attempts to maximize its own data rate either by transmitting at full power in IW, or by setting its corresponding weight wco to unity in OSB, ISB, and ASB. This produces the rate regions shown in Figure 5.14, which shows that ASB achieves near optimal performance similar to OSB and ISB, and significant gain over IW even though both ASB and IW are autonomous. For example, with a target rate of 1 Mbps on CO, the rate on RT3 reaches 7.3 Mbps under the ASB algorithm, which is a 121% increase compared with the 3.3 Mbps achieved by IW. We have also performed extensive simulations (more than 10, 000 scenarios) with different CO and RT positions, line lengths, and reference line parameters. We found that the performance of ASB is

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very insensitive to definition of the reference line: with a single choice of the reference line we observe good performance in a broad range of scenarios.

5.4.7 Concluding Remarks and Future Directions Dynamic spectrum management techniques can greatly improve the performance of DSL lines by inducing cooperation among interfering users in the same binder. For example, the iterative waterfiling algorithm is a completely autonomous DSM algorithm with linear complexity in the number of users and number of tones, but the performance could be far from optimal in the mixed CO/RT deployment scenario. The optimal spectrum balancing and iterative spectrum balancing algorithms achieve optimal and close to optimal performances, respectively, but have high complexities in terms of the number of users and are completely centralized. This section surveys an autonomous dynamic spectrum management algorithm called autonomous spectrum balancing. ASB utilizes the concept of “reference line”, which mimics a typical victim line in the binder. By setting the power spectrum level to protect the reference line, a good balance between selfish and global maximizations can be achieved. Compared with IW, OSB, and ISB, the ASB algorithm enjoys completely autonomous operations, low (linear) complexity in both the number of users and number of tones. Simulation shows that the ASB algorithm achieves close to optimal performance and is robust to the choice of reference line parameters.


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We conclude this section by highlighting the key ideas behind ASB. The reference line represents the statistical average of all victims within a typical network, and can be thought as a “static pricing”. This differentiates the ASB algorithm with power control algorithms in the wireless setting, where pricing mechanisms have to be adaptive to the change of channel fading states and network topology, or Internet congestion control, where time-varying congestion pricing signals are used to align selfish interests for social welfare maximization. By using static pricing, no explicit message passing among the users is needed and the algorithm becomes autonomous across the users. This is possible because of the static nature of the channel gains in DSL networks. Mathematically, the surprisingly good rate region results by ASB means that the specific engineering problem structures in this nonconvex and coupled optimization problem can be leveraged to provide a very effective approximation solution algorithm. Furthermore, robustness of the attained rate region with respect to perturbation of the reference line parameters has been verified to be very strong. This means that the dependence of the values of the local maxima of this nonconvex optimization problem on crosstalk channel coefficients is sufficiently insensitive for the observed robustness to hold. There are several exciting further directions to pursue with ASB, for example, convergence conditions for ASB-I, extensions to intercarrier-interference cases, and bounds on optimality gap that are empirically verified to be very small. Interactions of ASB with link layer scheduling have resulted in further improvement of throughput in DSL networks [33, 67].

5.5 Internet Routing 5.5.1 Introduction Most large IP (Internet protocol) networks run interior gateway protocols (IGPs) such as OSPF (open shortest path first) or IS-IS (intermediate system—intermediate system) that select paths based on link weights. Routers use these protocols to exchange link weights and construct a complete view of the topology inside the autonomous system (AS). Then, each router computes shortest paths (where the length of a path is the sum of the weights on the links) and creates a table that controls the forwarding of each IP packet to the next hop in its route. To handle the presence of multiple shortest paths, in practice, a router typically splits traffic roughly evenly over each of the outgoing links along a shortest path to the destination. The link weights are typically configured by the network operators or automated management systems, through centralized computation, to satisfy traffic-engineering goals, such as minimizing the maximum link utilization or the sum of link cost [24]. Following common practice, we use the the sum of some increasing and convex

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link cost functions as the primary comparison metric and the optimization objective in this section. Setting link weights under OSPF and IS-IS can be categorized as linkweight-based traffic engineering, where a set of link weights can uniquely and distributively determine the flow of traffic within the network for any given traffic matrix. The traffic matrix can be computed based on traffic measurements (e.g., [20]) or may represent explicit subscriptions or reservations from users. Link-weight-based traffic engineering has two key components: a centralized approach for setting the routing parameters (i.e., link weights) and a distributed way of using these link weights to decide the routes to forward packets. Setting the routing parameters based on a networkwide view of the topology and traffic, rather than the local views at each router, can achieve better performance [22]. Evaluation of various traffic engineering schemes, in terms of total link cost minimization, can be made against the performance benchmark of optimal routing (OPT), which can direct traffic along any paths in any proportion. The formulation can be found, for example, in [70]. OPT models an idealized routing scheme that can establish one or more explicit paths between every pair of nodes, and distribute an arbitrary amount of traffic on each of the paths. It is easy to construct examples where OSPF, one of the most prevalent IP routing protocols today, with the best link weighting performs substantially (5000 times) worse than OPT in terms of minimizing sum of link cost. In addition, finding the best link weights under OSPF is NP-hard [24]. Although the best OSPF link weights can be found by solving an integer linear program (ILP) formulation, such an approach is impractical even for a midsize network. Many heuristics, including local search [24] and simulated annealing [5, 18] have been proposed to search for the best link weights under OSPF. Among them, local-search technique is the most attractive method in finding a good setting of the link weights for large-scale networks. Even though OSPF with a good setting of the weights performs within a few percent of OPT for some practical scenarios [24, 18, 5], there are still many realistic situations where the performance gap between OSPF and OPT could be significant even at low utilization. There are two main reasons for the difficulty in tuning OSPF for good performance. First, the routing mechanism restricts the traffic to be routed only on shortest paths (and evenly split across shortest paths, an issue that has been addressed in [59]). Second, link weights and the traffic matrix are not integrated into the optimization formulation. Both bottlenecks are overcome in the distributed exponentially weighted flow splitting (DEFT) protocol developed in [70]: 1. Traffic is allowed to be routed on nonshortest paths, with exponential penalty on path lengths.


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2. An innovative optimization formulation is proposed, where both link weights and flows are variables. It leads to an effective two-stage iterative method. As a result, DEFT, discussed in this section, has the following desirable properties. • It determines a unique flow of traffic for a given link weight setting in polynomial time. • It is provably always better than OSPF in terms of minimizing the maximum link utilization or the sum of link cost. • It is readily implemented as an extension to the existing IGP (e.g., OSPF). • The traffic engineering under DEFT with the two-stage iterative method realizes near-optimal flow of traffic even for large-scale network topologies. • The optimizing procedure for DEFT converges much faster than that for OSPF. In summary, DEFT provides a new way to compute link weights for OSPF that exceeds the current benchmark based on local search methods while reducing computational complexity at the same time. Furthermore, the performance turns out to be very close to the much more complicated and difficult to implement family of MPLS-type routing protocols, which allows arbitrary flow splitting. More recently in [71], we have proved that a variation of DEFT, called PEFT, can provably achieve the optimal traffic engineering as a link-state routing protocol with hop-by-hop forwarding, with the optimal link weights computed in polynomial time and much faster than local search methods for link weight computation for OSPF. This answers the question on optimal traffic engineering by link state routing conclusively and positively.

5.5.2 DEFT: Framework and Properties Given a directed graph G = (V, E) with capacity cu,v for each link (u, v), let D(s, t) denote the traffic demand originated from node s and destined for node t. Φ(fu,v , cu,v ) is a strictly increasing convex function of flow fu,v on link (u, v), typically a piecewise linear cost [24,P59] as shown in equation (5.45). The networkwide objective is to minimize (u,v)∈E Φ(fu,v , cu,v ). ⎧ fu,v ⎪ ⎪ ⎪ ⎪ 3f ⎪ u,v − 2/3 cu,v ⎪ ⎨ 10fu,v − 16/3 cu,v Φ(fu,v , cu,v ) = ⎪ 70fu,v − 178/3 cu,v ⎪ ⎪ ⎪ 500fu,v − 1468/3 cu,v ⎪ ⎪ ⎩ 5000fu,v − 16318/3 cu,v

fu,v /cu,v ≤ 1/3 1/3 ≤ fu,v /cu,v ≤ 2/3 2/3 ≤ fu,v /cu,v ≤ 9/10 (5.45) 9/10 ≤ fu,v /cu,v ≤ 1 1 ≤ fu,v /cu,v ≤ 11/10 11/10 ≤ fu,v /cu,v .

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In link-weight-based traffic engineering, each router u needs to make an independent decision on how to split the traffic destined for node t among its outgoing links only using link weights. Therefore, it calls for a function (Γ (·) ≥ 0) to represent the traffic allocation. Shortest path routing (e.g., OSPF) evenly splits flow across all the outgoing links as long as they are on the shortest paths. First of all, we need a variable to indicate whether link (u, v) is on the shortest path to t. Denote wu,v as the weight for link (u, v), and dtu as the shortest distance from node u to node t; then dtv + wu,v is the distance from u to t when routed through v. The gap of the two above distances, htu,v = dtv + wu,v − dtu is always larger than or equal to 0. Then (u, v) is on the shortest path to t if and only if htu,v = 0. Accordingly, we can use a unit step function of htu,v to represent the traffic allocation for OSPF as follows. ½ 1, if htu,v = 0 t Γ (hu,v ) = (5.46) 0, if htu,v > 0. The flow proportion on the outgoing link (u, v) destined for t at u is X Γ (htu,j ). Γ (htu,v )/ (u,j)∈E

t Denote fu,v as the flow on link (u, v) destined for node t and fut as the flow sent along the shortest path of node u destined for t; then t fu,v = fut Γ (htu,v ).

(5.47)

The Γ (htu,v ) function (5.46) (i.e., evenly splitting) results in intractability in searching for the best link weights under OSPF. In part inspired by Fong et al.’s work in [21], we can define a new Γ (htu,v ) function to allow for flow on nonshortest paths. Intuitively, we may want to send more traffic on the shortest path than on a nonshortest path. Moreover, the traffic on a nonshortest path should be 0 if the distance gap between the nonshortest path and the shortest path is infinitely large. Based on the above intuition, Γ (htu,v ) should be a strictly decreasing continuous function of htu,v bounded within [0, 1]. The exponential function is one of the natural choices, and the performance of using such function turns out to be excellent. In [70], we propose an IGP with distributed exponentially weighted flow splitting: ½ −ht e u,v , if dtu > dtv t Γ (hu,v ) = (5.48) 0, otherwise; that is, the routers can direct traffic on nonshortest paths, with an exponential penalty on longer paths. The following properties of DEFT can be proved [70].


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Proposition 5.6. DEFT can realize any acyclic flow for a single-destination demand within polynomial time. It can also achieve optimal routing with a single destination within polynomial time. For any traffic matrix, it can determine a unique flow for a given link weighting within polynomial time. Proposition 5.7. DEFT is always better than OSPF in terms of minimizing total link cost or the maximum link utilization.

5.5.3 DEFT: Optimization Formulation and Solutions Note that it is still difficult to directly integrate the exponentially weighted flow splitting of DEFT into an optimization formulation because of its discrete feature; that is, the traffic destined for node t can be sent through link (u, v) if and only if dtu > dtv . Instead of introducing some binary variables, we relax (5.48) into (5.49) first, and then, by properly setting the lower bound of all link weights, a constant parameter wmin , make such relaxation as tight as we want: t Γ (htu,v ) = e−hu,v . (5.49) Indeed, consider a flow solution satisfying (5.49); there is a link (u, v) where t t t t t dtv ≥ dtu and fu,v > 0, then fu,v ≤ fut e−hu,v = fut e−(dv +wu,v −du ) ≤ fut e−wmin . If wmin is large enough, this flow portion, which is infeasible to DEFT on link (u, v), could be neglected. Therefore, we present the following optimization problem, called ORIG, using the relaxed rule of flow splitting as the approximation for the traffic engineering under DEFT. P minimize Φ(fu,v , cu,v ) (5.50) subject to

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(5.52) (5.53) (5.54) (5.55) (5.56)

Note that both the flow splittings and the link weights are incorporated as optimization variables in one problem, with further constraints relating them. Constraint (5.51) is to ensure flow conservation at an intermediate node y. Constraint (5.52) is for flow aggregation on each link. Constraint (5.53) is from the definition of gap of shortest distance. Constraints (5.54) and (5.55) come from (5.47) and (5.49). In addition, (5.54) and (5.55) also imply that

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t fu,v ≤ fut , and that htu,v of at least one of an outgoing links (u, v) of node u destined for node t should be 0; that is, the link (u, v) is on the shortest path from node u to node t. Problem ORIG is nonsmooth and nonconvex due to nonsmooth constraint (5.55) and nonlinear equality (5.54). In [70], we propose a two-stage iterative relaxation to solve problem ORIG. First, we relax constraint (5.55) into (5.57) below: X t fut ≤ fu,v , ∀t ∈ V, ∀u ∈ V. (5.57) (u,v)∈E

Equations (5.50)—(5.54), (5.56), and (5.57) constitute problem APPROX. We only need to obtain a “reasonably” accurate solution (link weighting W) to problem APPROX because the inaccuracy caused by the relaxation (5.57) will be compensated by a successive refinery process later. From the W, we can derive the shortest path tree T(W, t)6 for each destination t, and t all other dependent variables (dtu , htu,v , fut , fu,v , fu,v ) within DEFT. We then use these values as the initial point (which is also strictly feasible) for a new problem REFINE, which consists of equations (5.50)—(5.54), (5.56), and (5.58) below: t fut = fu,v ,

∀t ∈ V ∩ ∀u ∈ V ∩ (u, v) ∈ T(W, t).

(5.58)

With the two-stage iterative method, we are left with two optimization problems, APPROX and REFINE, both of which have convex objective functions and twice continuously differentiable constraints. To solve the largescale nonlinear problems APPROX and REFINE (with O(|V ||E|) variables and constraints), we extend the primal—dual interior point filter line search algorithm, IPOPT [68], by solving a set of barrier problems for a decreasing sequences of barrier parameters μ converging to 0. In summary, in solving problem APPROX, we mainly want to determine the shortest path tree for each destination (i.e., deciding which outgoing link should be chosen on the shortest path). Then in solving problem REFINE, we can tune the link weights (and the corresponding flow) with the same shortest path trees as in APPROX. The pseudocode of the proposed two-stage iterative method for DEFT is shown in Algorithms 6A and 6B. Most instructions are self-explanatory. Function DEFT FLOW(W) is used to derive a flow from a set of link weights W. Given the initial and ending values for barrier parameter μ, maximum iteration number, with/without initial link weighting/flow, function DEFT IPOPT() returns a new set of link weights as well as a new flow. Note that, as shown in Algorithm 6B, when DEFT IPOPT() is used for problem APPROX, it returns with the last iteration rather than the iteration with the best Flowi in terms of the objective value as in problem REFINE. This 6 To keep T(W, t) as a tree, only one downstream node is chosen if a node can reach the destination through several downstream nodes with the same distance.


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is because problem APPROX has different constraints from problem ORIG and a too greedy method may leave small search freedom for the successive REFINE problem. Finally, we need to specify initial and terminative μ values, (μinit ≥ μend approx ≥ μend refine ), and maximum iteration number Iterapprox ≥ Iterrefine . As shown in the next section, it is straightforward to specify these parameters. Algorithm 6A. DEFT Solution. 1. 2. 3. 4.

(μ, W) ← DEFT IPOPT(μinit , μend approx , Iterapprox , nil) Initial Point ← (W, DEFT FLOW(W)) (μ, W) ← DEFT IPOPT(μ, μend refine , Iterrefine , Initial Point) Return (W, DEFT FLOW(W))

Algorithm 6B. DEFT IPOPT. If Initial Point 6= nil Then Initiate the problem with Initial Point /*REFINE*/ End if For each iteration i ≤ Itermax with μstart ≥ μ ≥ μend do μi ← current value for μ Wi ← current values for all wu,v Flowi ← DEFT FLOW(Wi ) end for If Initial Point = nil then return (μi , Wi ) of the last iteration /*APPROX*/ else return (μi , Wi ) of the iteration with the best Flowi in terms of objective value /*REFINE*/ end if

5.5.4 Numerical Examples We summarize some of the numerical results in [70] on various schemes under many practical scenarios. We employ the same cost function (5.45) as in [23]. The primary metric used is the optimality gap, in terms of total link cost, compared against the value achieved by optimal routing using CPLEX 9.1 [16] via AMPL [25]. The secondary metric used is the maximum link utilization. We do not reproduce the performance of some obvious link-weight-based traffic engineering approaches for OSPF, for example, UnitOSPF (setting all link weights to 1), RandomOSPF (choosing the weights randomly), InvCapOSPF (setting the weight of an link inversely proportional to its capacity as recommended by Cisco), or L2OSPF (setting the weight proportional to its physical Euclidean distance) [23], because none of them performs as well as

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the state-of-the-art local search method proposed in [23]. In addition, because DEFT is always better than OSPF in terms of minimizing the maximum link utilization or the sum of link cost, we bypass the scenarios where OSPF can achieve near-optimal solution. Instead, we are particularly interested in those scenarios where OSPF does not perform well. For fair comparisons, we use the same topology and traffic matrix as those in [23]. The 2-level hierarchical networks were generated using GTITM, which consists of two kinds of links: local access links with 200-unit capacity and long distance links with 1000-unit capacity. In the second type of topology, the random topologies, the probability of having a link between two nodes is a constant parameter and all link capacities are 1000 units. Although AT&T’s proprietary code of local search used in [23] is not publicly available, there is an open source software project with IGP weight optimization, TOTEM 1.1 [66]. It follows the same lines as [23], and has similar quality of the results. It is slightly slower due to the lack of implementation of the dynamic Dijkstra algorithm. We use the same parameter setting for local search as in [24, 23] where link weight is restricted as an integer from 1 to 20, initial link weights are chosen randomly, and the best result is collected after 5000 iterations. To implement the proposed two-stage iterative method for DEFT, we modify another open source software, IPOPT 3.1 [34], and adjust its AMPL interface to integrate it into our test environment. We choose μinit = 0.1 for most cases except for μinit = 10 for the 100-node network with heavy traffic load. We also choose μend approx = 10−4 , μend refine = 10−9 , and maximum iteration number Iterapprox = 1000, Iterrefine = 400. The code terminates earlier if the optimality gap has been less than 0.1%. Example 5.9. DEFT and OSPF on 2-level topology. The results for a 2-level topology with 50 nodes and 212 links with seven different traffic matrices are shown in Table 5.5. The results are also depicted graphically in Figure 5.15.

Table 5.5 Results of 2-level topology with 50 nodes and 212 links Total Traffic Demand Ave Link Load-OPT Max Link Load-OPT Opt. Gap-OSPF Opt. Gap-DEFT

1700 0.128 0.667 2.8% 0.1%

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3100 0.242 0.9 64.2% 0.1%

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In addition to the two metrics, optimality gap in terms of total link cost and maximum link utilization,7 we also show the average link utilization under optimal routing as an indication of network load. From the results, we 7 Note, however, maximum link utilization is not a metric as comprehensive as total link cost because it cannot indicate whether there are multiple overcongested links.

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can observe that the gap between OSPF and optimal routing can be very significant (up to 222.8%) for a practical network scenario, even when the average link utilization is low (≤27%). In contrast, DEFT can achieve almost the same performance as the optimal routing in terms of both total link cost and maximum link utilization. Example 5.10. DEFT and OSPF on random topology. Similar observations can be found for other scenarios, for example, as shown in Figure 5.17. Without exception, the curves of the DEFT scheme (the horizontal lines almost coinciding with x-axes) almost completely overlap those of optimal routing,

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in terms of total link cost and maximum link utilization. Among these numerical experiments, the maximum optimality gap of OSPF is as high as 252% and that of DEFT is only at worst 1.5%. In addition, DEFT reduces the maximum link utilization compared to OSPF on all tests, and substantially on some tests. Simulations on rate of convergence, as well as comparisons of computation and implementation complexity, can be found in [70].

5.5.5 Concluding Remarks and Future Directions Network operators today try to alleviate congestion in their own network by tuning the parameters in IGP. Unfortunately, traffic engineering under OSPF or IS-IS to avoid networkwide congestion is computationally intractable, forcing the use of local-search techniques. While staying within the context of link-weight-based traffic engineering, we propose a new protocol called [70] distributed exponentially weighted flow splitting. DEFT significantly outperforms the state-of-the-art OSPF local search mechanisms in minimizing networkwide congestion. The success of DEFT can be attributed to two additional features. First, DEFT can put traffic on nonshortest paths, with an exponential penalty on longer paths. Second, DEFT solves the resulting optimization problem by integrating link weights and the corresponding traffic distribution together in the formulation. The novel formulation leads to a much more efficient way of tuning link-weight than the existing local search heuristic for OSPF.


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DEFT is readily implementable as an extension to existing IGPs. It is provably always better than OSPF in minimizing the sum of link cost. DEFT retains the simplicity of having routers compute paths based on configurable link weights, while approaching the performance of the much more complex routing protocols that can split traffic arbitrarily over any paths. In summary, in terms of minimizing total link cost, performance of OSPF by local search heuristics is at best what is attained by solving the ILP, which is substantially outperformed by DEFT that comes very close to the optimal routing. In terms of a performance—complexity tradeoff, DEFT clearly exceeds OSPF. In this section, we only address the link weighting under DEFT for a given traffic matrix. The next challenge would be to explore robust optimization under DEFT, optimizing to select a single weight setting that works for a range of traffic matrices and/or a range of link/node failure scenarios. Extension of the ideas behind DEFT to routing across different autonomous systems managed by different network operators is another interesting future direction. In the larger picture of “design for optimizability”, DEFT shows one case where by changing the underlying protocol, the resulting new optimization formulation becomes much more readily solvable or approximable. We expect this new approach to tackle nonconvex problems to bring many new results and insights to the engineering of communication networks. Indeed, in an extension of DEFT work [71], we have developed the first provably optimal link state routing protocol with hop by hop forwarding, called PEFT, which achieves optimal traffic engineering with polynomial time (and very fast in practice) computation of optimal link weights. Acknowledgments The author would like to acknowledge collaborations with Raphael Cendrillon, Maryam Fazel, Prashanth Hande, Jianwei Huang, Daniel Palomar, Jennifer Rexford, Chee Wei Tan, and Dahai Xu while working on the five publications related to this survey [29, 19, 14, 7, 70], as well as very helpful discussions on these topics with Stephen Boyd, Rob Calderbank, John Doyle, David Gao, Jiayue He, David Julian, Jang-Won Lee, Ying Li, Steven Low, Marc Moonen, Daniel O’Neill, Asuman Ozdaglar, Pablo Parrilo, Ness Shroff, R. Srikant, Ao Tang, and Shengyu Zheng. The work reported in this chapter has been supported in part by the following grants: AFOSR FA9550-06-1-0297, DARPA W911NF-07-1-0057, NSF CNS-0519880 and CNS-0720570, and ONR N00014-07-1-0864.

References 1. M. Avriel, Ed., Advances in Geometric Programming, Plenum Press, New York, 1980. 2. N. Bambos, “Toward power-sensitive network architectures in wireless communications: Concepts, issues, and design aspects,” IEEE Pers. Commun. Mag., vol. 5, no. 3, pp. 50—59, 1998. 3. S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, “A tutorial on geometric programming,” Optim. Eng., vol. 8, no. 1, pp. 67—127, 2007. 4. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.

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Chapter 6

Multilevel (Hierarchical) Optimization: Complexity Issues, Optimality Conditions, Algorithms Altannar Chinchuluun, Panos M. Pardalos, and Hong-Xuan Huang

Summary. In this chapter we discuss some algorithmic and theoretical results on multilevel programming including complexity issues, optimality conditions, and algorithmic methods for solving multilevel programming problems. We also discuss an approach, which is called the multivariate partition approach, for solving a single-level mathematical programming problem based on its equivalent multilevel programming formulation. Key words: Hierarchy, multilevel programming, multivariate partition approach

6.1 Introduction The word hierarchy comes from the Greek word “ι ραρχια,” a system of graded (religious) authority. Hierarchical (multilevel) structures are found in many complex systems and in particular in biology. Biological systems are characterized by hierarchical architectural designs in which organization is controlled on length scales ranging from the molecular to macroscopic. These hierarchical architectures rely on critical interfaces that link structural elements of disparate scale. Nature makes very different systems (that have specific hierarchical composite structures) out of very similar molecular constituents. First, the structures are organized in discrete levels. Second, the levels of structural organizaAltannar Chinchuluun · Panos M. Pardalos Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, U.S.A., e-mail: [email protected], [email protected] Hong-Xuan Huang Department of Industrial Engineering, Tsinghua University e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_6, © Springer Science+Business Media, LLC 2009

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tion are held together by specific interactions between components. Finally, these interacting levels are organized into an oriented distinct hierarchical composite system of specific function. The mathematical study of hierarchical structures can be found in diverse scientific disciplines including environment, ecology, biology, chemical engineering, classification theory, databases, network design, game theory, and economics. The study of hierarchy occurring in biological structures reveals interesting properties as well as limitations due to different properties of molecules. Understanding the complexity of hierarchical designs requires systems methodologies that are amenable to modeling, analyzing, and optimizing these structures. Hierarchical optimization (or multilevel) can be used to study properties of these hierarchical designs. In hierarchical optimization, the constraint domain is implicitly determined by a series of optimization problems that must be solved in a predetermined sequence. Hierarchical optimization is a generalization of mathematical programming. The simplest two-level (or bilevel) programming problem describes a hierarchical system composed of two levels of decision makers. A bilevel programming problem consists of two optimization problems where the constraint set of the upper-level problem is implicitly determined by the lower-level problem. min F (x, y) x

(6.1)

s.t. G(x, y) ≤ 0, where y solves min f (x, z) z

s.t. g(x, z) ≤ 0.

x ∈ X ⊂ Rn , y, z ∈ Y ⊂ Rm , where X ⊆ Rn and Y ⊆ Rm are compact sets, and G : Rn × Rm → Rp , g : Rn × Rm → Rq , and F, f : Rn × Rm → R are scalar functions. Let Ω = {(x, y) | G(x, y) ≤ 0, g(x, y) ≤ 0, (x, y) ∈ Rn × Rm } be the constraint set of the problem. We also denote the set of solutions of the lower-level programming problem by S(x) for any x ∈ X. When the objective functions and constraint functions are linear, Problem (6.1) is called a bilevel linear programming problem. For a bilevel linear programming problem, it is known that the solution to the problem occurs at an extreme point of the feasible set [16, 9]. If at least one of the objective functions or constraint functions is nonlinear, then Problem (6.1) is called a nonlinear bilevel programming problem. In general, bilevel optimization problems are nonconvex, and therefore, it is not easy to find globally optimal solutions.

6 Multilevel (Hierarchical) Optimization

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Bilevel programming has a wide range of applications in many areas including economics [8], network design [82, 15], transport planning and modeling [69, 71], credit allocation [74], and electric power pricing [51]. The reader is referred to Du and Pardalos [38] and Migdalas et al. [72] for a more comprehensive survey of applications. An excellent bibliographical survey of bilevel programming can also be found in Vicente and Calamai [88]. It seems that hierarchical structures are harder to be managed than completely centralized systems. Then what are the rationalities for hierarchical structures to exist? Answers to such questions may help us to understand the reason behind hierarchical structures in biology. Multilevel programming problems have been studied extensively in their general setting for the last decade. Many algorithmic developments are based on the properties of special cases of the bilevel problem (and the more general problem) and reformulations to equivalent or approximating models, presumably more tractable. Most of the exact methods are based on branchand-bound or cutting plane techniques and can handle only moderately sized problems. The present chapter is organized as follows. In Section 6.2, we consider complexity issues of bilevel programming, and Section 6.3 covers optimality conditions of the problem. We discuss some of the algorithmic methods for solving the bilevel programming problem in Section 6.4. We present a multivariate partitional approach for solving single-level mathematical programming problem based on its equivalent bilevel programming formulation in Section 6.5. We also discuss two real-world applications of bilevel programming in transportation systems in Section 6.6, and some open problems in multilevel programming in Section 6.7.

6.2 Complexity Issues Complexity analyzes the intrinsic difficulty of optimization problems and reveals surprising connections among many other optimization problems and their solutions.

6.2.1 Complexity of Finding a Global Solution Calamai and Vicente [22] proposed a technique to generate bilevel programming problems. Using their technique, a linear bilevel programming problem, which has an exponential number of local minima, can be generated. Jeroslow [56] has shown that the linear bilevel program is N P-hard. His results were subsequently confirmed and simplified by Ben-Ayed and Blair [14] and Blair [19], and strengthened by Hansen et al. [46]. Hansen et al. demonstrated

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that the linear bilevel problem without the upper-level constraints, which is a more restricted max-min problem, is strongly N P-hard by considering the complexity of a special case of a linear max-min optimization problem. They reduced KERNEL to this problem whereas KERNEL is known to be N P-hard [43]. Max-min linear programs can be formulated as convex maximization (or concave minimization) problems. The computational complexity of min-max optimization problems has been extensively studied in Ko and Lin [61] These problems are naturally characterized by Π2P , the second level of polynomial-time hierarchy. For higher-level hierarchies, Jeroslow has shown that the optimal value of a (k + 1)-level linear program is Σk -hard.

6.2.2 Complexity of Local Search Methods Computing locally optimal solutions is presumably easier than finding globally optimal solutions in practice. However, from the complexity point of view it has been shown that the problem of checking local optimality for a feasible point and the problem of checking whether a local minimum is strict, are NP-hard even for instances of quadratic problems with a simple structure in the constraints and the objective. These results have been proved using the classical 3-satisfiability problem. Vicente et al. [90] proved that checking local optimality in bilevel programming is an N P-hard problem. The proof uses a similar idea as in Pardalos and Schnitger [77].

6.2.3 Approximation Algorithms Jeroslow [56] observed that for any constant factor c, it remains N P-hard to find a solution within a multiplicative factor of c of the optimum. Deng et al. [37] have extended this result to disallow even an additive constant for a sufficiently small multiplicative factor. That is, for sufficiently small c1 > 0 and for some c2 > 0, there is no algorithm that can guarantee a solution within (1 + c1 ) · optimum + c2 unless N P = P.

6.2.4 Polynomially Solvable Problems Liu and Spencer [65] have introduced a polynomial-time algorithm for a bilevel linear problem when there are a constant number of lower-level control variables.


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Deng et al. [37] have presented a much simpler proof for the above result which also allows for an extension to a k-level linear programming problem when the total number of variables controlled by lower-level linear programs is a constant.

6.3 Optimality Conditions Bard [9] first derived optimality conditions for the bilevel programming problem based on an equivalent single-level programming problem of the bilevel programming problem. However, Clark and Westerberg [30] showed a counterexample of these conditions. Ye et al. [95] presented necessary optimality conditions for the generalized bilevel programming problem, which is a bilevel programming problem with a variational inequality as the lower-level problem. They proved that the generalized bilevel programming problem is equivalent to single-level problems under some assumptions on the objective function. Then, Karush—Kuhn— Tucker optimality conditions are applied to the single-level problems to derive optimality conditions for the generalized bilevel programming problem. Let us define the concept of global and local optimality of the bilevel problem: min F (x, y)

(6.2)

x

s.t. G(x) ≤ 0, y solves min f (x, z) z

s.t. g(x, z) ≤ 0.

x ∈ Rn , y, z ∈ Rm , where X = {x | G(x) ≤ 0} is a compact set.

Definition 6.1. A point (x∗ , y ∗ ) is called a local optimal solution of Problem (6.2), if and only if, x∗ ∈ X, y ∗ ∈ S(x∗ ) with F (x∗ , y ∗ ) ≤ F (x∗ , y),

for all y ∈ S(x∗ ),

and there exists δ > 0 such that φ(x∗ ) ≤ φ(x) for all x ∈ X ∩ B(x∗ , δ), where φ(x) = min{F (x, y) | y ∈ S(x)}. y

The point (x∗ , y ∗ ) is called a global optimal solution of Problem (6.2) if we can select as δ = ∞.

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Here, B(x∗ , δ) is the open ball of radius δ centered at point x∗ ∈ X; that is, B(x∗ , δ) = {x ∈ Rn | kx−x∗ k < δ}. As we mentioned before, the bilevel problem can be formulated as a single-level problem by replacing the lower-level problem by its Karush—Kuhn—Tucker conditions under some constraint qualification if the lower level problem is convex. Then, the resulting problem is a smooth optimization problem. However, Scheel and Scholtes [80] showed that the equivalent single-level formulation of the bilevel problem violates most of the usual constraint qualifications. Therefore, they introduced a nonsmooth version of Karush—Kuhn—Tucker conditions: min F (x, y)

x,y,λ

(6.3)

s.t. G(x) ≤ 0, ∇y L(x, y, λ) = 0, min{−gj (x, y), λj } = 0, j = 1, . . . , q, where L(x, y, λ) = f (x, y) + λT g(x, y) is the Lagrangian of the lower-level problem. Then, every local solution of Problem (6.2) is a local solution of Problem (6.3). It is not difficult to see that every local solution (x∗ , y ∗ , λ∗ ) of Problem (6.3) is also a local solution of the problem min F (x, y)

x,y,λ

(6.4)

s.t. G(x) ≤ 0, ∇y L(x, y, λ) = 0, gj (x, y) = 0 for gj (x∗ , y ∗ ) = 0, λj = 0 for λ∗j = 0, gj (x, y) ≤ 0 for gj (x∗ , y ∗ ) < 0, λj ≥ 0 for λ∗j > 0. Let us consider the Mangasarian—Fromowitz constraint qualification [68]: Definition 6.2. We say that Mangasarian—Fromowitz constraint qualification (MFCQ) is satisfied at point (x, y) if there exists a vector d ∈ Rm satisfying dT ∇y gj (x, y) < 0 for all j ∈ I(x, y) = {i | gi (x, y) = 0, i = 1, . . . , q}. Now, we are ready to present a theorem [35], which is an application of the results in [80], for Problem (6.3). Theorem 6.1. Let (x∗ , y ∗ , λ∗ ) be a local minimum of Problem (6.3). Suppose that (MFCQ) is satisfied for Problem (6.4) at (x∗ , y ∗ , λ∗ ). Then, there exist multipliers (u, v, w, r) such that


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∇F (χ∗ ) + uT (∇x G(x∗ ), 0) + ∇(∇y L(χ∗ , λ∗ )v) + wT ∇g(χ∗ ) = 0, ∇y g(χ∗ )v − r = 0, gj (χ∗ )wj = 0, j = 1, . . . , q, λ∗j rj = 0, j = 1, . . . , q, wj rj ≥ 0, j ∈ {i | gi (x∗ , y ∗ ) = λ∗i = 0}, uT G(x∗ ) = 0, u ≥ 0, where χ∗ = (x∗ , y ∗ ). Vicente and Calamai [89] presented necessary and sufficient optimality conditions for bilevel programming problems with quadratic strictly convex lower-level problems using on the local geometry of the problems. Based on the geometrical property of the problems, they observed that the set of feasible directions at a point is a finite union of some convex sets, and extended the first- and second-order optimality conditions of single-level programming to bilevel programming.

6.4 Algorithms Algorithmic approaches for the bilevel problem may be grouped as follows: (1) extreme point ranking methods, (2) branch-and-bound algorithms, (3) complementarity pivot algorithms, (4) descent methods, (5) penalty function methods, and (6) trust region methods. In this section, we discuss some of these algorithmic approaches for solving the bilevel programming problem. In particularly, extreme point algorithms, branch-and-bound algorithms, and a multicriteria approach are covered. Complementarity pivoting is a method based on replacing the lower-level problem with its Karush—Kuhn—Tucker conditions. Many authors suggested different pivot algorithms for the modified problem including [18, 59, 73]. Descent methods [79, 90, 62, 40] look for a local solution of the bilevel problem based on feasible descent directions with respect to the upper-level function. Penalty function methods [4, 5, 60, 20, 93] are also local search methods for solving the bilevel problem.

6.4.1 Extreme Point Algorithms As we mentioned before, the linear bilevel programming problem has a nice property that the solution of the problem occurs at an extreme point of the feasible set. Therefore, the problem can be solved using a vertex enumeration technique. Candler and Townsley [24] proposed the first enumeration algo-

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rithm for the linear bilevel programming problem, which has no upper-level constraints and the lower-level problem with unique solution. Their algorithm enumerates the basis of the lower-level problem. One of the well-known vertex enumeration algorithms is the kth best method introduced by Bialas and Karwan [17]. The kth best method enumerates the basis of the resulting problem of the bilevel problem after relaxing the objective function of the lower-level problem. Tuy et al. [86] proposed a global optimization approach for the linear twolevel problem: min cTu x + dTu y x≥0

(6.5)

s.t. Au x + Bu y ≤ ru , where y solves min cTl x + dTl z z≥0

s.t. Al x + Bl z ≤ rl ≤ 0.

x ∈ Rn , y, z ∈ Rm .

We can choose cl = 0 because the value of this parameter does not affect the solution of the linear bilevel problem. Therefore, Problem (6.5) is equivalent to the following reverse convex programming problem. min cTu x + dTu y s.t. Ax + By ≤ r dTl y ≤ S(x) x, y ≥ 0, where A = (Au , Al )T , B = (Bu , Bl )T , r = (ru , rl )T , and S(x) is the optimal objective function value of the lower-level problem min dTl z s.t. Al x + Bl z ≤ rl y ≥ 0. The function S(x) is a convex polyhedral function and the problem is a linear program with an additional reverse convex constraint. Several algorithms [49, 83] are available for these types of problems. They applied the approach proposed by Tuy [84] to reduce the dimension of the above optimization problem. Then, a vertex enumeration technique is applied to solve the resulting problem. Zhang and Liu [96] proposed an extreme point algorithm to solve a mathematical programming problem with variational inequalities. As we mentioned


205

before, this problem generalizes the bilevel programming problem in which the lower-level problem is convex. Some other vertex enumeration methods can be found in [32, 26].

6.4.2 Branch-and-Bound Algorithms One way to tackle the bilevel problem is to replace the lower-level problem by its corresponding Karush—Kuhn—Tucker conditions and obtain a single-level mathematical programming problem. However, if the lower-level problem is nonconvex, the corresponding single-level problem is no longer equivalent to the bilevel problem. Most of the branch-and-bound algorithms for the bilevel programming problem are based on the single-level programming problem min F (x, y) s.t. G(x, y) ≤ 0, g(x, y) ≤ 0, q X ∇y f (x, y) + λj ∇y g(x, y) = 0, j=1

λj gj (x, y) = 0, λ ≥ 0, x ∈ X, y ∈ Y.

Due to the stationary and complementarity conditions, the above problem is nonconvex even if the bilevel problem is linear. Bard and Moore [12] implemented a branch-and-bound algorithm, which was initially suggested by Fortuny-Amat and McCarl [41], for solving the bilevel problem where the constraints and the objective function of the upperlevel problem are linear and, the objective function of the lower-level problem is quadratic. Hansen et al. [46] derived necessary optimality conditions for linear bilevel programming, expressed in terms of active constraints of the lower-level problem. Based on the optimality conditions, they established a branch-andbound algorithm for linear bilevel programming. Their computational results showed that this approach was favorable for the linear case compared to the methods by Bard and Moore [12] and J´ udice and Faustino [58]. Al-Khayyal et al. [6] and Bard [10] also introduced branch-and-bound algorithms for the quadratic bilevel problem. G¨ um¨ us and Floudas [45] proposed a branch-and-bound algorithm for the bilevel programming problem. Their approach is based on a relaxation, made by Karush—Kuhn—Tucker conditions of the feasible region of the bilevel problem. The relaxed problem is solved using the global optimization technique described in [2, 3]. When the objective and constraint functions are twice differentiable and the linear constraint qualification holds for the lower-level program constraints, this approach guarantees global optimality. Bard and

206


Moore [13] and Wen and Yang [92] have also proposed branch-and-bound algorithms for integer and mixed integer linear bilevel programming problems.

6.4.3 A Multicriteria Approach Several authors have been interested in the relationship between bicriteria programming and the bilevel programming problem including Bard [9] and ¨ u [87] for the linear case. They claimed that an optimal solution of the Unl¨ linear bilevel problem is a nondominated point for the objective functions of the upper-level and the lower-level programming problems. However, Candler [23], Clarke and Westerberg [30], and Haurie et al. [48] reported counterexamples of this argument. F¨ ulöp [42] pointed out that more than two criteria are needed to establish the relationship between multicriteria programming and bilevel programming. F¨ ulöp showed that, for each linear bilevel programming problem, there exists some linear multicriteria problem such that the global solution of the first problem is an optimal solution for minimizing the upperlevel objective function over the Pareto optimal set of the second problem. In order to discuss a multicriteria approach, we need to define the concept of domination. Definition 6.3. A point a in a set M is said to be nondominated with respect to an order “≺”, if there does not exist any point b ∈ M such that b ≺ a and a 6= b. Recently, Fliege and Vicente [39] established the link between these two programs. Let us consider the bilevel programming problem (6.1) with X = Rn and Y = Rm . They defined an order such that every nondominated point of Rn × Rm is a solution to the bilevel problem. Let (x1 , y1 ), (x2 , y2 ) ∈ Rn × Rm . Then the order “≺” can be defined as follows. Definition 6.4. x1 = (x1 , y1 ) ≺ x2 = (x2 , y2 ) is equivalent to the following conditions. — x1 = x2 and f (x1 , y1 ) < f (x2 , y2 ) or — k∇y f (x1 , y1 )k2 = 0 and F (x1 , y1 ) < F (x2 , y2 ). Then, they proved that a nondominated point of Rn × Rm , with respect to the order defined above, is a solution of the bilevel problem. Let us define the following function, ϕ : (x, y) → (x, F (x, y), f (x, y), k∇y f (x, y)k2 ). A cone K can be defined in the image space of ϕ, Rn × R × R × R, as


207

K = {(x, f1 , f2 , d) | (x = 0 and f2 > 0) or (f1 > 0 and d ≥ 0)}. Then, we can define the usual cone order “ N , set h := h/2 and j = 0. We find a feasible point y ∈ Rn using the procedure GreedyRandomSol (x, h). Step 2. If f (y) < f (x), set j := 0, x := y and V al := f (x). Otherwise, set j = j + 1. Step 3. Set k := k + 1, and go to Step 1. Procedure GreedyRandomSol can be interpreted as follows. Algorithm 3. GreedyRandomSol Input: f , l, b, h, α ∈ (0, 1), x. Step 1. For each i = 1, . . . , n, find a solution x0i of the problem min f (x1 , . . . , xn ) s.t. li ≤ xi ≤ ui .

(6.11)

Let gi = f (x1 , . . . , xi−1 , x0i , xi+1 , . . . , xn ) for i = 1, . . . , n. Step 2. Let min and max be the minimum and the maximum values of the set {g1 , . . . , gn }, respectively. Let S = ∅. For each i = 1, . . . , n, S := S ∪ {i} if gi ≤ min + α(max − min). Step 3. Select an index l from the set S randomly, and set y := x and yl := x0l . Problem (6.11) can be solved by discretizing the solution space. The mesh points can be chosen as li , li + h, li + 2h, . . . , li + mh, where m is the largest integer number such that li + mh ≤ ui . Then we choose the minimum of the objective function values at the mesh points. It is not difficult to see that GRASP is a special case of the multivariate partition approach. In this case, the partition is {41 , . . . , 4n }, where 4i = xi for all i = 1, . . . , n, and Di = [li , ui ] for all i = 1, . . . , n.

212


6.5.2 Applications of the Multivariate Partition Approach Lennard-Jones Problem The Lennard-Jones problem is one of the most challenging global optimization problems in molecular biophysics. The problem is to find such a structure of a protein, a cluster of N atoms, interacting via the Lennard-Jones (dimensionless nonquantized pair) potential : v(r) = r−12 − 2r−6 , that its energy E is (globally) minimal. Let the code PN = {x1 , . . . , xN } be the collection of centers of N atoms. Then the potential energy E is defined as follows. X E(x1 , . . . , xN ) = v(kxi − xj k), 1≤i<j≤N

where v(r) = r−12 − 2r−6 . Therefore, our optimization problem can be formulated as X min E(x1 , . . . , xN ) = v(kxi − xj k), 1≤i<j≤N

3

s.t. x1 , . . . , xN ∈ R .

Huang et al. [53] presented necessary optimality conditions for the Lennard— Jones problem and applied the multivariate partitional approach to the Lennard-Jones problem. The partition can be described by a center of each atom; that is, 4i = xi for i = 1, . . . , N . They obtained all the expected global minima of the problem with 2 ≤ N ≤ 56 using a quasi-Newton accelerating approach for the auxiliary problem (6.9).

Spherical Code Problem The spherical code problem has many applications including physics, molecular biology, and chemistry. Tammes’ problem is one of the well-known cases of the spherical code problem which is referred to distribute points on a unit sphere such that they maximize the minimum distance between any pair of points. In general, we distribute points on a unit sphere according to a certain generalized energy. Let a code PN = {x1 , . . . , xN } be a set of N points on a unit sphere in Rn . Then the s-energy associated with the spherical code PN is ⎧ P kxi − xj k−s if s 6= 0 ⎪ ⎨ i<j ³ ´ w(s, PN ) = P 1 ⎪ ln kxi −x if s = 0, ⎩ jk i<j

and the spherical code problem can be formulated mathematically as


213

min fs (x1 , . . . , xN ) s.t. xi ∈ S n = {x | kxk = 1, x ∈ Rn } f or i = 1, . . . , n, where fs (PN ) =

½

w(s, PN ) −w(s, PN )

if s ≥ 0, if s < 0.

In [54], Huang et al. used a multivariate partition approach for the spherical code problems. The partition of the variables is similar to that of the Lennard-Jones problem.

6.6 Applications Multilevel optimization has many applications in different fields including economics, transportation systems, engineering system design, environmental engineering, and mechanics. Here, we discuss two of these applications in transportation systems. The concentration of human activities in urban areas has given rise to congestion problems that create negative environmental or economical effects. Therefore, many researchers have developed efficient methods for road network design in order to improve transportation systems. The network design problem (NDP) determines a set of parameters that optimizes the road network. The model includes traffic signal control, traffic information provision, congestion charge, new transportation modes, and road expansion or deletion. The NDP is usually formulated as a bilevel programming problem. The upper-level part defines system design and the lower-level part defines travellers’ behavior. NDPs are classified into two categories: discrete and continuous variations. LeBlanc [63] formulated a bilevel programming problem for discrete NDP. Discrete models usually consider link or lane additions. Abdulaal and LeBlanc [1] later described continuous models of the NDP. Continuous models are concerned with network improvements that can be modeled as continuous variables including lane and lateral clearance changes and other enhancements. The NDP has also been investigated by Janson and Husaini [55], Magnanti and Wong [67], and Xiong and Schneider [94]. For a more comprehensive survey about the NDP, the reader is referred to Migdalas [71] and Cascetta [25]. Another application of bilevel programming is the signal setting problem (SSP) which maximizes network performance by optimizing traffic signals. The main difficulty of the problem arises from the existing interaction between network performance and users’ choices of the routes. Gartner et al. [44] formulated the SSP as a bilevel programming problem, where the upperlevel part represents the public network manager and the lower-level part represents users’ behavior.

214


Some other applications of multilevel optimization can be found in Migdalas [71], Migdalas et al. [72], and Bard [11].

6.7 Some Interesting Open Problems 6.7.1 Polynomially Solvable Problems As we mentioned before, the linear bilevel programming problem is known to be N P-hard. However, if the lower-level problem has a constant number of variables, the linear bilevel problem is polynomially solvable. Hence, a similar question may arise: whether we can find a polynomial-time algorithm for the linear bilevel problem in which the upper-level problem has a constant number of variables. Therefore, it would be interesting if we can find polynomially solvable special cases of the nonlinear bilevel programming problem.

6.7.2 The Relation Between Multilevel Programming and Multicriteria Programming In Section 6.4, we discussed a bicriteria approach by Fliege and Vicente [39] for solving bilevel problems. They introduced an order in the Euclidean space and showed that an optimal solution of the bilevel problem is a nondominated point with respect to the order. However, they did not exactly define any multicriteria problem. The problem of finding equivalent multicriteria programs of multilevel programs is still an open question. As we mentioned ¨ u [87], have been made before, several attempts, including Bard [7] and Unl¨ to discover the relation between bilevel linear programming and multicriteria programming. Later, Wen and Hsu [91] pointed out that the theorem in [87], which illustrates the relation between multilevel programming and multicriteria programming, is valid only under an additional constraint. Let us consider the linear bilevel programming problem min F (x, y) = cTu x + dTu y x

where y solves min f (x, z) = cTl x + dTl z z

s.t. Ax + Bz ≤ r ≤ 0.

x ∈ Rn , y, z ∈ Rm and the multicriteria programming problem

(6.12)


min (cTu x + dTu y, dTl z) s.t. Ax + Bz ≤ r.

215

(6.13)

Then, Wen and Hsu stated the following theorem. Theorem 6.3. If ∇F T ∇f¯ ≥ 0, where f¯ = f (0, y), then the optimal solution to the bilevel programming problem (6.12) is an efficient solution to the bicriteria programming problem (6.13). The condition in the above theorem restricts the use of bicriteria approaches to the linear bilevel problem. Thus, it would be useful if we can derive an equivalent multicriteria programming problem, which may have three or more objective functions, for the billevel problem (6.12) without any strong conditions. Therefore, it might be interesting if we state special cases of the nonlinear multilevel programming problem, which are equivalent to multicriteria programming problems.

6.7.3 Multilevel Multicriteria Programming Problems Most of the real-world optimization problems require more than one objective function. Therefore, one of the challenging problems in mathematical programming is the multilevel multicriteria programming problem; that is, each decision maker has several objectives. The general bilevel multicriteria programming problem has the form min F (x, y) = (F1 (x, y), . . . , Fl (x, y)) x

(6.14)

s.t. G(x, y) ≤ 0, where y is Pareto optimal for min f (x, z) = (f (x, z), . . . , fk (x, z)) z

s.t. g(x, z) ≤ 0.

x ∈ X ⊂ Rn , y, z ∈ Y ⊂ Rm , where X ⊆ Rn and Y ⊆ Rm are compact sets, G : Rn × Rm → Rp , g : Rn × Rm → Rq , and Fi , fj : Rn × Rm → R, i = 1, . . . , l and j = 1, . . . , k, are scalar functions. Definition 6.6. A point y ∗ ∈ Y with f (x, y) is called Pareto optimal for the lower-level multicriteria problem, if and only if there exists no point z ∈ Y such that fi (x, z) ≤ fi (y) for all i = 1, 2, . . . , k and fj (x, z) < fj (x, y) for at least one index j ∈ {1, 2, . . . , k}.

Solving the bilevel multicriteria problem (6.14) means we look for Pareto optimal soltutions of the problem (6.14):

216


Definition 6.7. A point (x∗ , y ∗ ) ∈ X × Y with F (x∗ , y ∗ ), where y ∗ is Pareto optimal for the lower-level problem, is called Pareto optimal for (6.14), if and only if there exists no point (x, y) ∈ X × Y , where y is Pareto optimal for the lower-level problem, such that Fi (x) ≤ Fi (x∗ ) for all i = 1, 2, . . . , l and Fj (x, y) < Fj (x∗ , y ∗ ) for at least one index j ∈ {1, 2, . . . , l}. To our knowledge, this problem has not been studied by many researchers. Recently, Sinha and Sinha [81] studied multilevel decentralized programming problems in which decision makers have absolute control over certain decision variables but some variables may be controlled by two or more decision makers. In this case, the problem has conflicting objectives, but the decision makers are placed in hierarchical order. We have briefly discussed optimality conditions for multilevel programming in Section 6.3. Many researchers also considered optimality conditions and duality results for multicriteria programming problems including Preda [78], Jeyakumar and Mond [57], Liang et al. [64], and Chinchuluun et al. [28]. Based on the optimality conditions and duality results of both multicriteria programming and multilevel programming, optimality duality of multilevel multiobjective programming can also be studied.

6.8 Concluding Remarks In this brief survey, we have shown a number of theoretical results for deterministic multilevel programming. These results include complexity issues, optimality conditions, and algorithmic approaches for solving multilevel programming problems. We have also presented a method, which is called the multivariate partition approach, for solving single-level mathematical programming problems based on their equivalent bilevel programming formulations. Some open questions have also been included at the end of the chapter. This survey is not comprehensive; we have not focused on stochastic issues and connection with decomposition methods. Acknowledgments The research of the first two authors is partially supported by NSF, Air Force, and CRDF grants. The research of the third author is partially supported by National Science Foundation of China under Project 10201017.

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Chapter 7

Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes Antoine Deza, Tam´ as Terlaky, and Yuriy Zinchenko

Summary. We consider a family of linear optimization problems over the n-dimensional Klee—Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2n − 2 sharp turns. This fact suggests that any feasible path-following interior-point method will take at least O(2n ) iterations to solve this problem, whereas in practice typically only a few iterations (e.g., 50) suffices to obtain a high-quality solution. Thus, the construction potentially exhibits the worst-case iteration-complexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n). Key words: Linear programming, central path, interior-point methods, total curvature

7.1 Introduction Consider the following linear programming problem: min cT x such that Ax ≥ b where A ∈ Rm×n , b ∈ Rm , and c, x ∈ Rn . In theory, the so-called feasible path-following interior-point methods exhibit polynomial iteration-complexity: starting at a point on the central path √ they take at most O( m ln ν) iterations to attain a ν-relative decrease in the duality gap. Moreover, if L is the bit-length of the input data, it takes Antoine Deza · Tam´ as Terlaky · Yuriy Zinchenko Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, ON L8S 4K1, Canada e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_7, © Springer Science+Business Media, LLC 2009

223

224

A. Deza, T. Terlaky, and Y. Zinchenko

√ at most O( mL) iterations to solve the problem exactly; see, for instance, [11]. However, in practice typically only a few iterations, usually less than 50, suffices to obtain a high-quality solution. This remarkable difference stands behind the tremendous success of interior-point methods in applications. Let ψ : [α, β] → Rn be a C 2 map with nonzero derivative ∀t ∈ [α, β]. Denote its arc length by Z t ˙ )kdτ, kψ(τ l(t) := α

its parametrization by the arc length by ψarc (l) : [0, l(β)] → Rn , and its curvature at the point l, d κ(l) := ψärc (l). dl The total curvature K is defined as Z l(β) kκ(l)kdl. K := 0

Intuitively, the total curvature is a measure of how far off a certain curve is from being a straight line. Thus, it has been hypothesized that the total curvature of the central path is positively correlated with the number of iterations that any Newton-like path following method will take to traverse this curve, in particular, the number of iterations for feasible path-following interior-point methods, for example, long-step or predictor-corrector. The worst-case behavior for path-following interior-point methods has already been under investigation, for example, Todd and Ye [13] gave a lower √ iteration-complexity bound of order 3 m necessary to guarantee a fixed decrease in the central path parameter and consequently in the duality gap. At the same time, different notions for the curvature of the central path have been examined. The relationship between the number of approximately straight segments of the central path introduced by Vavasis and Ye [14] and a certain curvature measure of the central path introduced by Sonnevend, Stoer, and Zhao [12] and further analyzed in [15], was further studied by Monteiro and Tsuchiya in [9]. Dedieu, Malajovich, and Shub [1] investigated a properly averaged total curvature of the central path. Nesterov and Todd [10] studied the Riemannian curvature of the central path in particular relevant to the so-called short-step methods. We follow a constructive approach originated in [4, 5] which is driven by the geometrical properties of the central path to address these questions. We consider a family of linear optimization problems over the n-dimensional Klee—Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2n − 2 sharp turns. We derive explicit formulae for the number of the redundant constraints needed. In particular, we give a bound of

7 Redundant Klee—Minty Cubes

225

O(n23n ) on the number of redundant constraints when the distances to those are chosen uniformly. When these distances are chosen to decay geometrically, we give a slightly tighter bound of the same order n3 22n as in [5]. The behavior of the central path suggests that any feasible path-following interior-point method will take at least order 2n iterations to solve this problem. Thus, the construction potentially exhibits the worst-case iterationcomplexity known to date which almost matches the theoretical iterationcomplexity bound for this type of methods. However, state-of-the art linear optimization solvers that include preprocessing of the problem as described in [6, 7] are expected to recognize and remove the redundant constraints in no more than two passes. This underlines the importance of the implementation of efficient preprocessing algorithms. We show that the total curvature of the central path for the construction is at least exponential in n and, therefore, provides a counterexample to a conjecture of Dedieu and Shub [2] that it can be bounded by O(n). Also, the construction may serve as an example where one can relate the total curvature and the number of iterations almost exactly. The chapter is organized as follows. In Section 7.2 we introduce a family of linear programming problems studied along with a set of sufficient conditions that ensure the desired behavior for the central path and give a lower bound on the total curvature of the central path, in Section 7.3 we outline the approach to determine the number of the redundant constraints required, and Sections 7.4 and 7.5 contain a detailed analysis of the two distinct models for the distances to the redundant constraints. We give a brief conclusion in Section 7.6.

7.2 Sufficient Conditions for Bending the Central Path and the Total Curvature Let x ∈ Rn . Consider the following optimization problem. ⎧ min xn ⎪ ⎪ ⎪ ⎪ 0 ≤ x1 ≤1 ⎪ ⎪ ⎪ ⎪ εx ≤ x ≤ 1 − εxk−1 k = 2, . . . , n ⎪ k−1 k ⎨ 0 ≤ d1 + x1 repeated h1 times ⎪ εx ≤ d + x repeated h2 times ⎪ 1 2 2 ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎩ repeated hn times. εxn−1 ≤ dn + xn

The feasible region is the Klee—Minty n-cube and is denoted by C ⊂ Rn . Denote d := (d1 , . . . , dn ) ∈ Rn+ — the vector containing the distances to the redundant constraints from C, h := (h1 , . . . , hn ) ∈ Nn — the vector containing the number of the redundant constraints.

226


By analogy with the unit cube [0, 1]n , we denote the vertices of C as follows. For S ⊂ {1, . . . , n}, a vertex v S of C satisfies ½ 1 if 1 ∈ S v1S = 0 otherwise ½ S 1 − εvk−1 if k ∈ S vkS = k = 2, . . . , n. S εvk−1 otherwise Define δ-neighborhood Nδ (vS ) of a vertex v S , with the convention x0 = 0, by ½ ½ ¾ 1 − xk − εxk−1 ≤ εk−1 δ if k ∈ S S Nδ (v ) := x ∈ C : k = 1, . . . , n . otherwise xk − εxk−1 ≤ εk−1 δ Remark 7.1. Observe that ∀S ⊆ {1, . . . , n} for Nδ (vS ) to be pairwisedisjoint it suffices ε + δ < 1/2: given ε, δ > 0, the shortest amongst all n coordinates’ distance between the neighborhoods, equal to (1 − 2ε − 2εδ), is attained along the second coordinate and must be positive, which is readily implied. For brevity of the notation we introduce slack variables corresponding to the constraints in the problem above as follows: s1 sk s¯1 s¯k s˜1 s˜k

= x1 = xk − εxk−1 = 1 − x1 = 1 − εxk−1 − xk = d1 + x1 = dk + (xn − εxn−1 )

k = 2, . . . , n k = 2, . . . , n k = 2, . . . , n.

Recall that the analytic center χ corresponds to the unique maximizer arg max x

n X (ln si + ln s¯i + hi ln s˜i ). i=1

Also, recall that the primal central path P can be characterized as the closure of the set of maximizers ( ) n X n x ∈ R : x = arg max (ln si + ln s¯i + hi ln s˜i ), for some α ∈ (0, χn ) . x:xn =α

i=1

P Therefore, setting to 0 the derivatives of ni=1 (ln si + ln s¯i + hi ln s˜i ) with respect to xn , 1 hn 1 − + = 0, (7.1) sn s¯n sñ

and with respect to xk ,


227

ε 1 ε hk εhk+1 1 − − − + − = 0, sk sk+1 s¯k s¯k+1 s˜k s˜k+1

k = 1, . . . , n − 1,

(7.2)

combined give us necessary and sufficient conditions for x = χ. Furthermore, (7.2) combined with xn = α ∈ (0, χn ) gives us necessary and sufficient conditions for x ∈ P \ ({0} ∪ {χ}) where 0 ∈ Rn denotes the origin. Given ε, δ > 0, the sufficient conditions for h = h(d, ε, δ) to guarantee that the central path P visits the (disjoint) δ-neighborhoods of each vertex of C may be summarized in the following proposition. We write 1 for the vector of all ones in Rn . Proposition 7.1. Fix ε, δ > 0. Denote for k = 2, . . . , n Iδk := {x ∈ C : s¯k ≥ εk−1 δ, sk ≥ εk−1 δ} and Bδk := {x ∈ C : s¯k−1 ≤ εk−2 δ, sk−2 ≤ εk−3 δ, . . . , s1 ≤ δ}.

If h = h(d, ε, δ) ∈ Nn satisfies

Ah ≥

3 1 δ

(7.3)

and hk k−1 hk+1 k 3 ≥ ε + , ε dk + 1 dk+1 δ where

then

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ A=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 d1 +1 −1 d1

.. . −1 d1

−ε d2

0

−ε2 2ε d2 +1 d3

.. .

..

.

.. .

0 .. .

··· .. .

−1 d1

0

−1 d1

0

k = 1, . . . , n − 1,

0

···

0

0

0 ..

··· .. .

0 .. .

0 .. .

··· .. .

0 .. .

.

k−1

−εk 2ε dk +1 dk+1

.. .

..

0

0

···

0

0

···

.

2εn−2 −εn−1 dn−1 +1 dn

0

2εn−1 dn +1

(7.4)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Iδk ∩ P ⊂ Bδk . Proof. Fix k ≥ 2 and let x ∈ Iδk ∩ P. Let j ≤ k − 2. Summing up all of the ith equations of (7.2) over i = j, . . . , (k − 2), each multiplied by εi−1 , and then subtracting the (k − 1) st equation multiplied by εk−2 , we have

228

−


2hk−1 εk−2 hj εj−1 hk εk−1 εj−1 εk−1 εk−1 + + + + + s˜k−1 s˜j s˜k sj sk s¯k =

k−3 X εi 2εk−2 εj−1 + +2 . sk−1 s¯j s¯ i=j i+1

Because s˜k−1 < dk−1 + 1, s˜k > dk , s˜j > dj , and sk ≥ εk−1 δ, s¯k ≥ εk−1 δ as x ∈ Iδk , from the above we get εj−1 2 hj εj−1 hk εk−1 2hk−1 εk−2 − ≤ + . − dk−1 + 1 dj dk sj δ ¡ j−1 ¢ From (7.4) it follows that (h1 /d1 ) ≥ hj ε /dj , thus we can write

εj−1 2 h1 2hk−1 εk−2 hk εk−1 + ≤ + ; − d1 dk−1 + 1 dk sj δ ¡¡ ¢ ¡ ¢ ¢ that is, as (3/δ) ≤ − (h1 /d1 ) + 2hk−1 εk−2 / (dk−1 + 1) − hk εk−1 /dk by (7.3), we have −

snj ≤ εj−1 δ,

∀j ≤ k − 2.

In turn, the (k − 1) st equation of (7.2), hk−1 εk−2 hk εk−1 εk−2 εk−1 εk−1 εk−2 − = + + − s˜k−1 s˜k s¯k−1 sk s¯k sk−1 implies εk−2 2 hk−1 εk−2 hk εk−1 ≤ + − dk−1 + 1 dk s¯k−1 δ ¡¡ ¢ ¡ ¢ ¢ and since (3/δ) ≤ hk−1 εk−2 / (dk−1 + 1) − hk εk−1 /dk by (7.4), we have s¯k−1 ≤ εk−2 δ. t u Proposition 7.2. Fix ε, δ > 0. If h ∈ Nn satisfies (7.3) and (7.4), then χ ∈ Nδ (v{n} ). Proof. Summing up all of the ith equations of (7.2) over i = k, . . . , (n − 1), each multiplied by εi−1 , and then subtracting (7.1) multiplied by εn−1 , we have n−2

X εi εk−1 hk εk−1 2εn−1 2hn εn−1 εk−1 − + − −2 − =0 sk s¯k s˜k sn s¯i+1 sñ i=k

implying 2hn εn−1 hk εk−1 εk−1 − ≤ . sñ s˜k sk


229

¡ ¢ Because sñ ≤ dn + 1, s˜k ≥ dk and by (7.4), (h1 /d1 ) ≥ hk εk−1 /dk , from the above we get εk−1 2hn εn−1 h1 ≤ , − dn + 1 d1 sk ¢ ¢ ¡¡ combined with 2hn εn−1 / (dn + 1) − (h1 /d1 ) ≥ (3/δ) (from (7.3)) this leads to εk−1 sk ≤ δ k = 1, . . . , n − 1. 3 ¡ ¢ ¡ ¢ In turn, (7.1) hn εn−1¢/˜ sn ≤ εn−1 /¯ sn . And because ¡ implies ¡ n−1sñ 0 such that ε + δ < 1/2. If h ∈ Nn satisfies (7.3) and (7.4), then the central path P intersects the disjoint δ-neighborhoods of all the vertices of C. Moreover, ´ P ís confined to a polyhedral tube defined by Sn ³³Tn j c Tδ := k=1 ∩ Bδk with the convention Bδ1 = C. j=k+1 (Iδ )

Remark 7.2. Observe that T0 is the sequence of connected edges of C starting from v {n} and terminating at v ∅ , and is precisely the path followed by the simplex method on the original Klee—Minty problem as it pivots along the edges of C. For simplicity of the notation we write T instead of Tδ when the choice of δ is clear. For a fixed δ, we define a turn of T adjacent to a vertex v S , or corresponding to Nδ (v S ) if the δ-neighborhoods are disjoint because in the latter case Nδ (vS ) determines v S uniquely, to be the angle between the two edges of C that belong to T0 and connect at this vertex. Intuitively, if a smooth curve is confined to a narrow tube that makes a sharp turn, then the curve itself must at least make a similar turn and thus have a total curvature bounded away from zero. It might be worthwhile to substantiate this intuition with a proposition. Proposition 7.3. Let Ψ : [0, T ] → R2 be C 2 , parameterized by its arc length t, such that Ψ ([0, T ]) ⊂ {(x, y) : 0 ≤ x ≤ a + b, 0 ≤ y ≤ b} ∪ {(x, y) : a ≤ x ≤ a + b, −a ≤ y ≤ b} and Ψ (0) ∈ {0} × [0, b], Ψ (T ¡ ) ∈ [a, a +¢b] × {−a}. Then the total curvature K of Ψ satisfies K ≥ arcsin 1 − 2b2 /a2 .

Proof. By the mean-value theorem, for any τ such that Ψ1 (τ ) = a we have τ ≥ a; recall that kΨ˙ k = 1. Thus, by the same theorem, ∃t1 such that |Ψ2 (t1 )| ≤ b/a. Similarly, ∃t2 such that |Ψ1 (t2 )| ≤ b/a. Now map the values of the derivative of Ψ at t1 and t2 onto a sphere and recall that the total curvature K between these two points corresponds to the length of a connecting curve on the sphere, thus bounded below by the length of the geodesic (which in this case is the same as the angular distance). A simple calculation completes the proof. t u

230

A. Deza, T. Terlaky, and Y. Zinchenko Ψ˙ 2

Ψ2 b (0, 0)

a

Ψ1

Ψ˙ 1

Ψ˙ (t1 )

Ψ˙ (t2 )

Fig. 7.1 Total curvature and geodesics.

Remark 7.3. Note that if b/a → 0, then the corresponding lower bound on the total curvature K approaches π/2. Next we construct a simple bound on the total curvature of P by picking suitable d, ε, and finally δ small enough, together with h, that results in a “narrow” polyhedral tube T . For X ⊆ Rn denote its orthogonal projection onto a linear subspace spanned by a subset S ⊆ {1, . . . , n} of coordinates, with coordinates corresponding to S c suppressed, by XS . For x, z ∈ Rn we denote (x, z) the straight line segment connecting the point x and z. Corollary 7.2. Fix n ≥ 2. If di = (n − 1)2n−i+2 , i = 1, . . . , n, ε=

n−1 , 2n

1 δ= 32n2 and h satisfies

µ ¶n−2 4 , 5

⎢⎛ ⎞ ⎥ ⎢ ⎥ n X ⎢ 3 e⎥ δ ⎠ ⎣ ⎝ |aij | h= 1 + max h⎦ , 3 i j=1 δ

where Ae h = 1, then the total curvature of the central path P satisfies 1 K≥ 2n

µ ¶n−2 8 . 5


231

X3 P

v {3} 1

v {2,3} v{1,3}

v {1,2,3}

v {2}

v∅

1

X2 v {1,2} 1

X1

v{1} (v{2} ){1,2}

= (y {2>3} ){1>2} X2

(v{1,2} ){1,2}

1

P{1,2}

¡

%

% 0

(v ∅ ){1,2}

Iδ2

¢

= (y {1>2>3} ){1>2}

(v{1} ){1,2} 0

¡

{1,2}

Bδ2

¢

{1,2}

= (y {1>3} ){1>2} 1

X1

= (y {3} ){1>2}

Fig. 7.2 Planar projection of the central path for n = 3.

Proof. That ε, δ, h = h(d, ε, δ) above satisfies the conditions of Corollary 7.1 and thus P is confined to the polyhedral tube T is established in Section 7.5. Instead of analyzing P ∈ Rn directly we derive the lower bound on the total curvature of P based on its planar projection P{1,2} . From Iδk ∩ P ⊂ Bδk , k = 2, . . . , n, it follows that P{1,2} will traverse the two-dimensional Klee—Minty cube C{1,2} at least 2n−2 times, every time originating in either Nδ (v ∅ ){1,2} or Nδ (v {2} ){1,2} and terminating in the other neighborhood, while confined to the polyhedral tube T{1,2} = s1 ≤ δ} ∪ {¯ s2 ≤ εδ}) ∩ C{1,2} . Thus, P{1,2} will make at least ({s2 ≤ εδ} ∪ {¯ 2n−1 “sharp turns”, each corresponding to a turn in Nδ (v {1,2} ){1,2} or Nδ (v {1} ){1,2} .

232


In order to understand how the turns of P{1,2} contribute to the total curvature of P we need the following lemma. Lemma 7.1. Let u b, vb ∈ R3 and u = (b u{1,2} , 0), v = (b v{1,2} , 0). If the angle := π − arccos arg

min

w∈span{b b u,b v },

w∈span{u,v}, kwk=kwk=1 b

w bT w

between the hyperplane spanned by u b, vb and the hyperplane spanned by u, v does not exceed arcsin ε, then the angle α b between u b and vb satisfies ¡ ¢ ¡ ¢ α α cos α − ε2 1−cos cos α + ε2 1+cos 2 2 ¡ ¢ ≤ cos α ¡ ¢ b≤ α α 1 + ε2 1−cos 1 + ε2 1+cos 2 2 where α is the angle between u and v.

Proof. Without loss of generality we may assume kuk = kvk = 1 with q q α α 1−cos α , v = − sin = − , u1 = sin α2 = 1−cos 1 2 2 2 q q α α u2 = cos α2 = 1+cos , v2 = cos α2 = 1+cos , 2 2

and, assuming that the angle is precisely arcsin ε, parameterize span{b u, vb} by span{u, v} and z = (z1 , z2 , 0) such that kzk = 1, writing x ∈ span{b u, vb} as x = (x1 , x2 , xT{1,2} z{1,2} ε). Introducing β such that z1 = cos β and z2 = sin β we have Ãr ! r ³ π α´ 1 − cos α 1 + cos α u b= , , ε cos β − + , 2 2 2 2 vb =

Ã r −

1 − cos α , 2

r

! ³ 1 + cos α π α´ , ε cos β − + , 2 2 2

and, therefore, ¡ ¢ ¡ ¢ cos α + ε2 cos β − π2 + α2 cos β − π2 − α2 u bT vb cos α b= =q ¡ ¢q ¡ ¢. kb ukkb vk 1 + ε2 cos2 β − π2 + α2 1 + ε2 cos2 β − π2 − α2

Denoting γ := β − (π/2) and differentiating the above with respect to γ we get (cos α b)0γ =

where

(1 + ε2 )(−32ε2 sin 2γ + 16ε2 sin(2γ + 2α) + 16ε2 sin(2γ − 2α)) , D


233

¡ D = 16 + 8ε2 cos(2γ − α) + 16ε2 + 8ε2 cos(2γ + α) + 2ε4 cos 2α ¢3/2 + 2ε4 cos 4γ + 4ε4 cos(2γ + α) + 4ε4 cos(2γ − α) + 4ε4 .

Setting the derivative to 0 and simplifying the numerator we obtain the necessary condition for the extremum of cos α b, 32ε2 (1 + ε2 ) sin 2γ(cos 2α − 1) = 0.

That is, γ = k (π/2) for k = 0, ±1, ±2, and so on. In particular, it follows that the minimum of cos α b is attained at βmin = 0 and the maximum is attained at βmax = π/2. The bounds are obtained by further substituting the critical values of β into the expression for cos α b and observing the monotonicity with respect to ε. t u

Although the full-dimensional tube T might make quite wide turns, the projected tube T{1,2} is bound to make the same sharp turn equal to ¡ ¡ √ ¢¢ (π/2) + arcsin ε/ 1 + ε2 each time T passes through the δ-neighborhood of a vertex v S , 1 ∈ S (e.g., consider the turn adjacent to v {1,3} for n = 3). For a moment, equip C and T with a superscript n ¯ to indicate the dimension of the cube, that is, the largest number of linearly independent vectors in span({v S : vS ∈ C n¯ }). Recalling the C n defining constraints, namely εxn−1 ≤ xn ≤ 1 − εxn−1 , we note that by construction of the Klee—Minty cube, whenever we increase the dimension from n ¯ to n ¯ + 1, C n¯ is affinely n ¯ +1 n ¯ +1 transformed into “top” and “bottom” n ¯ -dimensional faces Ftop and Fbottom n ¯ +1 of C ; that is, ⎛ ⎞ 0 µ ¶ ⎜ .. ⎟ I ⎜ ⎟ n ¯ +1 = C n¯ + ⎜ . ⎟ , Ftop (0, . . . , 0, −ε) ⎝0⎠ 1 µ ¶ I n ¯ +1 Fbottom = C n¯ , (0, . . . , 0, ε)

n ¯ +1 and where I is the identity n ¯ ×n ¯ matrix, and C n¯ +1 is the convex hull of Ftop n ¯ +1 Fbottom . Consequently, any two-dimensional space spanned by two connected n ¯ +1 n ¯ +1 edges of C n¯ +1 from T0n¯ +1 ∩ Ftop or T0n¯ +1 ∩ Fbottom is obtained by tilting the two-dimensional space spanned by the two corresponding edges of C n¯ from T0n¯ , lifted to Rn¯ +1 by the n + 1) st coordinate to zero, by an √ ¡ setting ¢ (¯ angle not exceeding arcsin ε/ 1 + ε2 , and moreover, not exceeding arcsin ε. Therefore, we are in position to apply Lemma 7.1 to bound how fast the cosine of a turn αS of T n adjacent to any v S ∈ C n with 1 ∈ S may approach its two boundary values of 1 or −1 by induction on the dimension n. Fixing n = 3, S ⊆ {1, 2, 3} such that 1 ∈ S, adding and subtracting 1 to cos αS we get

234


1 + cos αS ≥ and 1 − cos αS ≥

1 + cos αS{1,2} ³ ´ S{1,2} 1 + ε2 1−cos α2 1 − cos αS{1,2} ³ ´. S{1,2} 1 + ε2 1+cos α2

Furthermore, for any n ≥ 3 and v S with 1 ∈ S we can write 1 + cos αS ≥ 1 − cos αS ≥

1 + cos αS{1,2} (1 +

n−2 ε2 )

1 − cos αS{1,2}

≥

1−ε

n−2 , (1 + ε2 ) √ ¢ ¡ 1 + 2ε/ 5 ≥ , (1 + ε2 )n−2

(1 + ε2 )n−2 √ √ recalling −2ε/ 5 ≥ cos αS{1,2} = −ε/ 1 + ε2 ≥ −ε because ε ≥ 2. Observe that by construction ³S of a polyhedral ´tube T , a single linearly S connected component of T \ may be uniquely idenS⊆{1,...,n} Nδ (v )

tified with an edge (v R , v S ), R, S ⊆ {1, . . . , n}, of C from T0 by having a nonempty intersection with this component and thus we denote such a component by L(vR ,vS ) and refer to it as a section of T corresponding to R S S (v p , v ). Moreover, recalling the definition of Nδ (v ) and T , and noting that 2 2 3 δ + (εδ) + (εδ) + · · · ≤ δ + εδ + · · · ≤ 2δ because ε ≤ 1/2, we get that within a given section of a tube L(vR ,vS ) the Euclidean distance from ∀x ∈ L(vR ,vS ) to the compact closure of (v R , v S ) ∩ L(vR ,vS ) is bounded from above by 2δ. Let us consider what happens to the central path in the proximity of a vertex v S ∈ C such that 1 ∈ S. We do so by manufacturing a surrogate for a part of T that is easier to analyze. Fix v S ∈ C with 1 ∈ S and denote the two adjacent vertices to which vS is connected by the two edges from T0 by v R and v Q . Without loss of generality we may assume that R v{1,2} = (0, 1), S v{1,2} = (1, 1 − ε), Q v{1,2} = (1, ε), and vnR > vnS > vnQ , so that the central path P enters the part of the polyhedral tube T sectioned between these three vertices via Nδ (vR ) and exits via Nδ (vQ ). Define four auxiliary points x, z ∈ (v R , v S ) and x, z ∈ (v S , vQ ) satisfying x{1,2} z {1,2} x{1,2} z {1,2}

= (1 − 3δ, 1 − ε + 3εδ) + = (1 − 3δ, 1 − ε + 3εδ), = (1, 1 − ε − 3δ), = (1, 1/2).

1/2−ε−3δ √ (−1, ε), 1+ε2


235

X2

1

R v{1,2}

1%

T {1,2}

x{1,2}

z {1,2}

S v{1,2}

x{1,2}

P{1,2} 1@2

z {1,2}

T {1,2}

Q v{1,2}

%

% 0

X1 0

1 1

Fig. 7.3 Schematic drawing for the cylindrical tube segments.

Because the distance from any point to the (part of the) identifying edge of L(vR ,vS ) or L(vS ,vQ ) is no greater than 2δ and because (·){1,2} corresponds to the orthogonal projection from Rn onto its first two coordinates, we can define two cylindrical tube segments: T := {x ∈ Rn : minz∈(x,z) kx − zk ≤ 2δ} ∩ {x ∈ Rn : (x − z)T x ≤ (x − z)T x} ∩ {x ∈ Rn : (z − x)T x ≤ (z − x)T z} and

T := {x ∈ Rn : minz∈(x,z) kx − zk ≤ 2δ} ∩ {x ∈ Rn : (x − z)T x ≤ (x − z)T x} ∩ {x ∈ Rn : (z − x)T x ≤ (z − x)T z}

such that T ⊃ L(vR ,vS ) ∩ {x ∈ Rn : (x − z)T x ≤ (x − z)T x, (z − x)T x ≤ (z − x)T z}, T ⊃ L(vS ,vQ ) ∩ {x ∈ Rn : (x − z)T x ≤ (x − z)T x, (z − x)T x ≤ (z − x)T z}, and

¡ ¢ ¡ ¢ T ∩ Nδ (v R ) ∪ Nδ (v S ) = T ∩ Nδ (v S ) ∪ Nδ (v Q ) = ∅.

Therefore, P will traverse T and T , first entering T through its face corresponding to (x − z)T x = (x − z)T x and exiting through the face cor-

236


responding to (z − x)T x = (z − z)T z, and then entering T at a point with (x−z)T x = (x−z)T x and exiting through a point with (z −x)T x = (z −x)T z. Now we choose a new system of orthogonal coordinates in Rn that allows us apply the argument similar to that of Proposition 7.3 as follows. Let the first two coordinates correspond to the linear subspace spanned by (x, z) and (x, z); align the second coordinate axis with the vector (z, x), so that the vector (x, z) forms the same angle equal to αS with the second coordinate axis as with (x, z). Choose the rest (n − 2) coordinates so that they form an orthogonal basis for Rn . Consider parameterization of P by its arc length, Parc . Because the shortest distance between the two parallel faces of T that correspond to {x ∈ Rn : (x − z)T x = (x − z)T x} and {x ∈ Rn : (z − x)T x = (z − x)T z} is equal to k(x, z)k = 1/2 − ε − 3δ, by the mean-value theorem it takes at least (1/2 − ε − 3δ) change of the arc length parameter for Parc to traverse T . Noting that while traversing the tube T the second coordinate of Parc might change at most by 2 · |2δ sin αS | + |(1/2 − ε − 3δ) cos αS |, by the same theorem we deduce that ∃t1 such that ¯³ ´ ¯ 2|2δ sin αS | + |(1/2 − ε − 3δ) cos αS | ¯ ¯ ˙ ¯ Parc (t1 ) ¯ ≤ 1/2 − ε − 3δ 2 4δ ≤ | cos αS | + . 1/2 − ε − 3δ

Analogously, considering T along the ith coordinate with i 6= 2 we conclude that ∀i 6= 2, ∃ti such that ¯³ ´¯ ¯ ¯ ˙ ¯ Parc (ti ) ¯ ≤ i

4δ . 1/2 − ε − 3δ

We use the points t1 , t2 , . . . , tn to compute a lower bound on the total curvature contribution of a turn of P next to v S : recalling kP˙ arc k = 1, the total curvature of the part of P that passes through T and T (i.e., resulting from a turn of T adjacent to v S ) may be bounded below by the length of the shortest curve on a unit n-sphere that connects points P˙ arc (t1 ), P˙ arc (t2 ), . . . , P˙ arc (tn ) in any order. For simplicity, the latter length may be further bounded below by KS :=

min

4δ , |xjj |≤ 1/2−ε−3δ

≥

max

dist(x1 , xj )

max

kx1 − xj k,

j≥2 xi ∈Rn , i=1,...,n: kxi k=1, ∀i, 4δ , |x11 |≤| cos αS |+ 1/2−ε−3δ j≥2

min

j≥2 xi ∈Rn , i=1,...,n: kx1 k=1, 4δ |x11 |≤| cos αS |+ 1/2−ε−3δ , 4δ , |xjj |≤ 1/2−ε−3δ

j≥2


237

where dist(x, z) is the length of the shortest curve on a unit sphere between points x and z, that is, the geodesic. Clearly, the critical value for the last expression is attained, in particular, at xi ∈ Rn+ , ∀i, when kx1 k = 1, kx1 − xj k = kx1 − xi k, i, j ≥ 2, and x11 = | cos αS | +

4δ , 1/2 − ε − 3δ

xjj =

4δ , 1/2 − ε − 3δ

j ≥ 2.

It follows that µ n X 1 2 (xj ) = 1 − | cos αS | + j=2

4δ 1/2 − ε − 3δ

¶2

≥ 1 − | cos αS | −

4δ 1/2 − ε − 3δ

³ ¢n−2 ´ ¡ , and, because | cos αS | ≤ 1 − (1 − ε) / 1 + ε2 n X (x1j )2 ≥ j=2

1−ε 4δ − , (1 + ε2 )n−2 1/2 − ε − 3δ

resulting in ¶ 4δ 1−ε − (1 + ε2 )n−2 1/2 − ε − 3δ µ ¶ 4δ 1 1 1 − ≥ · , n − 1 2(1 + 1/4)n−2 n − 1 1/2 − ε − 3δ

x1j ≥ (x1j )2 ≥

1 n−1

µ

j ≥ 2.

¢ ¡ Therefore, recalling ε = (n − 1) /2n and δ = 1/32n2 (4/5)n−2 , we can write KS ≥ kx1 − x2 k ≥ x12 − x22

µ ¶n−2 µ ¶ 1 4 4δ − 1+ 5 n − 1 1/2 − ε − 3δ µ ¶n−2 µ ¶n−2 1 n 4 4 ≥ − 2 1 2(n − 1) 5 8n (n − 1) 5 − ≥

1 2(n − 1)

2n

µ ¶n−2 µ ¶n−2 4 4 1 n ≥ − 2 2n 2(n − 1) 5 8n (n − 1) 5 µ ¶n−2 1 4 ≥ . 4n 5

1 3 32n2

¡ 4 ¢n−2 5

Finally, recalling that the polyhedral tube T makes 2n−1 such turns, we conclude that the total curvature of P indeed satisfies K ≥ (1/2n) (8/5)n−2 . u t

238


The bound on the total curvature K of P established above is obviously not tight. We expect the true order of K to be 2n up to a multiplier, rational in n. Remark 7.4. In R2 , by combining the optimality conditions (7.1) and (7.2) for the analytic center χ with that of the central path P visiting the δneighborhoods of the vertices v {1} and v {1,2} one can show that for δ below a certain threshold both d1 and d2 are bounded away from 0 by a constant. In turn, this implies that for fixed feasible d1 , d2 , the necessary conditions (7.1) and (7.2) for h chosen such that the central path visits the δ-neighborhoods of all the vertices of C are “asymptotically equivalent” as δ ↓ 0 to the sufficient conditions (7.3) and (7.4), up to a constant multiplier. Here the term asymptotic equivalence refers to the convergence of the normalized extreme rays and the vertices of the unbounded polyhedra given by the set of necessary conditions for a fixed d to those of the polyhedra given by the set of sufficient conditions (7.3) and (7.4). This suggests that the following might be true. In Rn min di ≥ dˆ > 0,

i=1,...,n

where dˆ is independent of n, δ, ε. Moreover, the necessary conditions for P to visit the δ-neighborhoods of all the vertices of C for a fixed d are asymptotically equivalent as δ ↓ 0 to the sufficient conditions (7.3) and (7.4). If, furthermore, we confine ourselves to only bounded subsets of all such feasible (d, h) corresponding to, say, n X i=1

hi ≤ Hδ∗ := 2 min d,h

n X

hi

i=1

then the conditions (7.3) and (7.4) are tight, in a sense that if we denote the set of all (d, h) satisfying the necessary P conditions for P to visit all the δ-neighborhoods intersected with {h : i h ≤ Hδ∗ } asPNeccδ , the set of all (d, h) satisfying (7.3) and (7.4) intersected with {h : i h ≤ Hδ∗ } as Suff δ , then for some small enough δˆ there exists M, m > 0 independent of ε such that ˆ Neccmδ ⊆ Suff δ ⊆ NeccMδ , 0 < δ ≤ δ.

7.3 Finding h ∈ Nn Satisfying Conditions (7.3) and (7.4) We write f (n) ≈ g(n) for f (n), g(n) : N → R if ∃c, C > 0 such that cf (n) ≤ g(n) ≤ Cf (n), ∀n; the argument n is usually omitted from the notation.


239

Denote b := 1 (3/δ). Let us first concentrate on finding h ∈ Nn such that (7.3) holds. If the integrality condition on h is relaxed, a solution to (7.3) can be found by simply solving Ah = b. Note that kAk1,∞ = max i

n X j=1

|aij |

is, in fact, small for d — large componentwise and ε < 1/2. So to find an integral h we can • Solve Ab h = (1 + γ)b for some small γ > 0. • Set h = bb hc.

Observe that for h to satisfy (7.3), it is enough to require maxi (A(b h − h) − γb)i ≤ 0. In turn, this can be satisfied by choosing γ > 0 such that γ

n X 3 |aij |. ≥ max i δ j=1

In Section 7.3.1 we show how to solve this system of linear equations. In Section 7.3.2 we demonstrate that under some assumption on d, (7.4) is already implied by (7.3), and consequently the rounding of b h will not cause a problem for (7.4) either. Remark 7.5. The choice of rounding down instead of rounding up is arbitrary.

7.3.1 Solving the Linear System Because (1 + γ)b = (1 + γ) (3/δ) 1, we can first solve Ae h = 1 and then scale e h by (1 + γ) (3/δ). Our current goal is to find the solution to Ae h = 1. For an arbitrary invertible B ∈ Rn×n and y, z ∈ Rn such that 1 + z T B −1 y 6= 0, the solution to (B + yz T )x = b can be written as x = B −1 (b − αy), where α=

xT B −1 b 1 + z T B −1 y

(for writing (B + yz T )x = Bx + y(z T x) = b, denoting α := z T x, we can express x = B −1 (b − αy) and substitute this x into (B + yz T )x = b again to compute α).

240


Denoting ⎛

1 d1 +1

⎜ 0 ⎜ ⎜ . ⎜ . ⎜ . ⎜ B := ⎜ ⎜ 0 ⎜ . ⎜ .. ⎜ ⎜ ⎝ 0 0 y T :=

−ε 0 d2 −ε2 2ε d2 +1 d3

.. . 0 .. . 0

. ··· .. . 0

0

0

0 0 .. .

..

··· ··· .. .

0 0 .. . ··· .. .

2εk−1 −εk dk +1 dk+1

.. . 0

0

..

. ··· ···

0 0 .. . 0 .. .

2εn−2 −εn−1 dn−1 +1 dn 2εn−1 0 dn +1

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−1 (0, 1, 1, . . . , 1), d1

z T := (1, 0, 0, . . . , 0), h ≡ Ae h = 1 as we can compute the solution to (B + yz T )e ⎛ ⎞ 0 ⎟ ⎜ α ⎜1⎟ e h = B −1 1 + B −1 ⎜ . ⎟ , ⎝ .. ⎠ d1 1

(7.5)

where

α=

Pn

−1 B1j Pn −1 . j=2 B1j

j=1

1−

1 d1

(7.6)

So to get the explicit formula for e h we need to compute B −1 and P show−1that d can be chosen such that α is well defined; that is, 1 − (1/d1 ) nj=2 B1j 6= 0. In order to invert B first note that it satisfies µ ¶ 1 2ε 2ε2 2εn−1 B = Diag , , ..., (I + S), d1 + 1 d2 + 1 d3 + 1 dn + 1 where a superdiagonal matrix S ∈ Rn×n is such that ( −ε(di +1) j = i + 1, i = 1, . . . , n − 1 di+1 Sij = 0 otherwise. Recall that (I + Z)−1 = I − Z + Z 2 − Z 3 + · · · for any Z ∈ Rn×n such that these matrix-power series converge. In our case, the powers of S are easy to compute for 1 ≤ k ≤ n − 1,

7 Redundant Klee—Minty Cubes k Sij =

( Qi+k−1 l=i

241

Sl,l+1

0

j = i + k, i = 1, . . . , n − k otherwise,

and S m = 0 for all m ≥ n, so the inverse of (I + S) can be computed as above and the inverse of B can be further computed by post-multiplying by the inverse of µ ¶ 1 2ε 2ε2 2εn−1 Diag , , ..., . d1 + 1 d2 + 1 d3 + 1 dn + 1 Therefore, B −1 is equal to ⎛ ⎜ d1 + 1 ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ . ⎜ .. ⎜ ⎜ ⎝ 0

(d1 +1)(d2 +1) (d1 +1)(d2 +1)(d3 +1) (d1 +1)···(d4 +1) 2d2 4d2 d3 8d2 d3 d4

···

d2 +1 2ε

(d2 +1)(d3 +1) 4d3 ε

(d2 +1)···(d4 +1) 8d3 d4 ε

···

0

d3 +1 2ε2

(d3 +1)(d4 +1) 4d4 ε2

···

0

0

d4 +1 2ε3

.. .

.. .

.. .

···

0

0

0

..

.

···

Qn

(dj +1) j=1Q 2n−1 n j=2 dj Qn

(dj +1) j=2Q n j=3 dj

2n−1 ε Qn

j +1) j=3 (d Qn j=4 dj

2n−2 ε2 Qn

+1) j=4 (d Qjn j=5 dj

2n−3 ε3

.. .

dn +1 2εn−1

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

7.3.2 Partial Implication for Sufficient Conditions Observe that in order for e h ∈ Rn+ , we must have α > 0 as in (7.6). For if not (i.e., if α < 0), then denoting ⎛ ⎞ 0 n ⎜1⎟ X ⎜ ⎟ −1 β1 := z T B −1 ⎜ . ⎟ = B1j ⎝ .. ⎠ j=2

1

and writing

α=

β1 + d1 + 1 1 − βd11

we must have β1 > d1 > 0. So β1 + d1 + 1 −α = >1 d1 β1 − d1

(7.7)

242


and from (7.5) it follows that if (α/d1 ) < −1, then e h2 , e h3 , . . . , e hn < 0. From now on we assume α > 0 (in Sections 7.4.1, 7.5.1 we show how to achieve this by choosing d appropriately). Note that in this case ¡(α/d1 ) ¢> 1. Suppose h ∈ Nn is such that (7.3) holds. If, furthermore, hi εi−1 /di is dominated by h1 /d1 for i = 1, . . . , n, then (7.3) already implies (7.4). Therefore, it is left to show that d can be chosen such that h = bb hc satisfies the domination condition above. For this to hold it suffices 3 δ (1

+ γ)e h1 − 1 ≥ d1

3 δ (1

+ γ)e hi + 1 i−1 ε , di

where e h solves Ae h = 1. The above is implied by d1 + 1 + β1 + d1

α d1 β1

−

1 6

≥

(1 +

βi α d1 ) εi−1

+

di

i = 2, . . . , n,

1 6 i−1

ε

,

i = 2, . . . , n

because γ > 0, δ < 1/2, where βi := εi−1 (B −1 1)i . This can be written as µ ¶ ¶ µ 5 α βi εi−1 α β1 +1+ ≥ 1+ + , 1+ d1 d1 6d1 d1 di 6di

i = 2, . . . , n.

In particular, if we have β1 βi > , d1 di

i = 2, . . . , n

(7.8)

then the above inequality holds true if 1≥

5 ε − , 6d1 6d1

that is, because ε < 1/2, for di > 0 for i = 1, . . . , n. Finally, observe that if d1 ≥ di , i ≥ 2, and d1 = O(2n ), then the magnitude of e h is primarily determined by α: recalling (7.5), (7.7), we write ⎛ ⎛ ⎞ ⎞ 0 ⎜ B −1 ⎜ 1 ⎟ B −1 ⎟ ⎜ ⎜ ⎟ ⎟ e h = α⎜ 1⎟ ⎜.⎟+ d1 ⎠ ⎝ d1 ⎝ .. ⎠ 1


243

⎛

⎛ ⎞ ⎞ ³ ´ 0 ⎜1⎟ 1 − βd11 B −1 ⎟ ⎜ ⎟ ⎟ + 1⎟ ⎜ .. ⎟ ⎝.⎠ d1 + 1 ⎠ 1

⎛

⎞⎞ ⎛ ⎞ ⎛ ⎛ ⎞ 0 d1 + 1 0 µ µ ¶ ¶ ⎟ ⎟⎟ ⎜1⎟ β1 β1 ⎜ B −1 ⎜ ⎜ ⎟ ⎜ 0 ⎟⎟ ⎜1⎟ ⎜ .. ⎟ + 1 − ⎜ .. ⎟⎟ . ⎜ .. ⎟ + 1 − ⎝.⎠ d1 d1 + 1 ⎝ . ⎠ d1 ⎝ . ⎠⎠ 1 1 0

⎜ B −1 ⎜ ≤ α⎜ ⎝ d1 ⎜ B −1 ⎜ = α⎜ ⎝ d1

Because di > 0 for i = 1, . . . , n, we have (di + 1) /2di > 12 and so 1 − (β1 /d1 ) < 1/2n−1 , implying ⎞⎞ ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ 0 d1 + 1 0 ⎟⎟ ⎜ B −1 ⎜ 1 ⎟ ⎜1⎟ B −1 1 ⎜ ⎜ ⎟ ⎜ 0 ⎟⎟ ⎜ ⎜ ⎟ e h < α⎜ ⎜ .. ⎟ + n−1 ⎜ .. ⎟ + n−1 ⎜ .. ⎟⎟ ⎝ . ⎠⎠ ⎝ d1 ⎝ . ⎠ 2 (d1 + 1) ⎝ . ⎠ 2 1 1 0 (7.9) ⎛ ⎞ 0 ⎟ B −1 ⎜ ⎜1⎟ ≈α ⎜ .. ⎟ for large n. d1 ⎝ . ⎠ 1

7.4 Uniform Distances to the Redundant Hyperplanes Clearly, many different choices for d are possible. In this section we explore the uniform model for d; that is d1 = d2 = · · · = dn .

7.4.1 Bounding α For α > 0 we need β1 < d1 . Denoting ξ :=

d1 + 1 2d1

the above can be written as (d1 + 1)ξ + (d1 + 1)ξ 2 + · · · + (d1 + 1)ξ n−1 < d1

244


or, equivalently, ξ 2 (1 + ξ + ξ 2 + · · · + ξ n−2 )
0, so p(ξ) > 0 for d1 large enough. Observe that ξ > 12 for any d1 > 0, and p0

¡1¢

2(n+1) 2n

0 p00 12 = −4 + 2(n+1)n n−1 2 ⎩ 0 2

so p

µ

1 + ∆ξ 2

¶

2

= −3 +

µ

1 ≥ pe + ∆ξ 2

¶

for n = 1, 2, . . . for n = 1 for n = 2, 3, 4 otherwise for k ≥ 3,

µ ¶ µ ¶ µ ¶ 1 1 ∆ξ 2 1 0 00 := p +p ∆ξ + p . 2 2 2 2

Thus, to guarantee p(ξ) > 0, it is enough to require µ ¶ 1 pe + ∆ξ ≥ 0 2

letting ξ = 12 + ∆ξ. Denoting

n(n + 1) 2n−1 2(n + 1) eb := −3 + 2n 1 e c := n 2

e a := −2 +

and

⎧ √ ⎪ ⎨ −eb+ eb2 −4eaec , 2e a ∆ξ ∗ := √ ⎪ ⎩ −eb− eb2 −4eaec , 2e a

2≤n≤4 n>4

(7.10)


245

(the smallest positive root of pe( 12 + ∆ξ) = 0), we conclude that β1 < d1 as long as ξ=

d1 + 1 1 ≤ + ∆ξ ∗ . 2d1 2

That is, d1 =

1 1 . ≥ 2∆ξ 2∆ξ ∗

(7.11)

Note that for d1 = d2 = · · · = dn , (7.8) holds, so (7.4) is readily implied by (7.3). It is left to demonstrate how to choose d1 satisfying the above to guarantee a moderate growth in e h as n → ∞.

7.4.2 Picking a “Good” d1 Note that as n → ∞, we have ∆ξ ∗ →

1 3 · 2n

by expanding the square root in (7.10) as a first-order Taylor series. Also, 1− and hence

β1 = p(ξ) ≥ pe(ξ) d1

1 1 ≤ . β1 p e (ξ) 1 − d1

In fact, for large n, p(ξ) ≈ pe(ξ), for 12 ≤ ξ ≤ linear on this interval because, as n → ∞, eb → −3 ←

1 2

e c ∆ξ ∗

+ ∆ξ ∗ . In turn, pe(ξ) is almost

(we compare the slope of pe( 12 + ∆ξ) at ∆ξ = 0 with its decrement as a function of ∆ξ over [0, ∆ξ ∗ ]). Recalling that the growth of e h is primarily determined by α for large n (see (7.9)), our goal becomes to minimize α. From (7.7), (7.11), noting that β1 ≈ d1 for large n (also recall β1 < d1 ), we get α≤

β1 + (d1 + 1) 2d1 2 1 ≈ ≈ · ∆ξ pe(ξ) pe(ξ) 2∆ξ ce − e c ∆ξ ∗

246


and, moreover the right-hand side of this expression approximates α fairly well. So, to approximately minimize e h, we maximize µ ¶ ∆ξ ∆ξ · e c−e c ∗ ∆ξ

for 0 ≤ ∆ξ ≤ ∆ξ ∗ , which corresponds to setting ∆ξ =

∆ξ ∗ 2

thus resulting in α → 6 · 22n and e hi = O

µ

22n εi−1

¶

as n → ∞ ,

i = 1, . . . , n.

7.4.3 An Explicit Formula For a given n, ε, δ, compute ∆ξ ∗ according to (7.10), set ∆ξ = ∆ξ ∗ /2, d1 as in (7.11). Compute the solution to Ae hP = 1 using (7.5), where α is computed according to (7.6). Set γ = (δ/3) maxi nj=1 |aij | and, finally, º ¹ 3e h = (1 + γ) h . δ

From (7.9) it follows that (for large n) ⎛ ⎞ 0 ⎟ ⎜ α α ⎜1⎟ e hi ≈ B −1 ⎜ . ⎟ ≤ i−1 , d1 ⎝ .. ⎠ ε 1

i = 1, . . . , n.

We are interested P in picking ε, δ, to minimize the total number of the redunn dant constraints, i=1 hi . Recalling ε + δ < 12 , denoting g(ε) :=

ε(ε−n − 1) 1 − 3ε + 2ε2

for large n we can write Pn

i=1

hi ≈

3 δ

Pn e i=1 hi ≈

Pn 3 i=1 δ 1 n 1− ( ) ε 6 22n 1−2ε 1− 1 ε 2n+2

≈ 6· = 9·2

g(ε)

α

¡ 1 ¢i−1 ε


247

(it is natural to pick δ as close to 12 − ε as possible, say δ = .999 In order to bound g ∗ := min1 g(ε)

¡1 2

¢ − ε ).

0 > 0, 1− n n(n + 1) 2 =

248


so g 0 (εU ) > 0 and thus g 0 (ε) > 0 for εU < ε < 1/2. Consequently, for ε ∈ (εL , εU ) we have µ ¶n −n(1 − 2ε) −2n3 8n 0 g (ε) ≥ min = n + 1 4n − 5 ε∈[εL ,εU ] εn (1 − ε)(1 − 2ε)2 and, therefore, by the mean-value theorem, g(εL ) +

min

ε∈[εL ,εU ]

g 0 (ε)(εU − εL ) ≤ g ∗ ≤ g(εL );

that is, 0
0.

7.5 Geometrically Decaying Distances to the Redundant Hyperplanes Next we explore the geometric model for d: di = ω (1/˜ ε)

n−i+1

, i = 1, . . . , n.

7.5.1 Bounding α As in Section 7.4.1, we need to guarantee β1 < d1 . Firstly, we give a lower bound on (∆k )k=1,...,n recursively defined by 1 − ∆k+1 = with ∆0 = 1, and where

d˜k+1 + 1 (2 − ∆k ), 2d˜k+1

k = 0, . . . , n − 1

(7.12)


d˜k = ω

249

µ ¶k 1 , ε˜

k = 1, . . . , n

with some constant ω. We have di = dñ−i+1 for i = 1, . . . , n, and β1 dñ + 1 = (1 − ∆n−1 ), d1 dñ

βi = 1 − ∆n−i+1 , di

i = 2, . . . , n.

Note that to satisfy β1 < d1 we necessarily must have 1 − ∆k < 1 for k = 1, . . . , n − 1. From (7.12) it follows that ∆k+1 =

∆k ε˜k+1 ∆k ε˜k+1 ∆k ε˜k+1 − + ≥ − , 2 ω 2 ω 2 ω

k = 1, . . . , n − 1

and hence ∆k ≥

∆1 ε˜2 − (1 + (2˜ ε) + (2˜ ε)2 + · · · + (2˜ ε)k−2 ), 2k−1 ω2k−2

Observing ∆1 =

1 2

k = 2, . . . , n.

(1 − (˜ ε/ω)) we can write the above inequality as

1 ∆k ≥ k 2

µ

1 = k 2

Ã

ε˜ 1− ω

¶

−

k−2 ε˜2 X (2˜ ε)i ω2k−2 i=0

k−2 ε˜ 4˜ ε2 X (2˜ ε)i 1− − ω ω i=0

!

,

k = 2, . . . , n.

Now, for α to be positive, that is, for µ ¶ 1 ∆n−1 dñ + 1 1− (1 − ∆n−1 ) = 1 − 1 − ∆n−1 + − > 0, dñ dñ dñ it suffices ∆n−1 −

1 >0 dñ

which is implied by 1 2n−1

Ã

n−3 ε˜ 4˜ ε2 X (2˜ ε)i 1− − ω ω i=0

!

−

If ε˜ = 12 , the above translates into 1 2n−1

−

n−1 > 0; ω · 2n−1

εñ > 0. ω

250


that is ω >n−1 resulting in di = ω2n−i+1 > (n − 1)2n−i+1 ,

i = 1, . . . , n.

It is left to verify that hi εi−1 /di is indeed dominated by h1 /d1 for i = 1, . . . , n, to ensure (7.4) as in Section 7.3.2. In particular, we demonstrate (7.8), as in the case of uniform d. Recalling β1 dñ + 1 = (1 − ∆n−1 ) d1 dñ and

βi = 1 − ∆n−i+1 di

for i = 2, . . . , n,

it immediately follows that β2 β1 > . d1 d2 Also observe β2 β3 βn β1 > > > ··· > d1 d2 d3 dn because, recalling (7.12) and 0 < ∆k < 1, k = 1, . . . , n − 1, ¶ µ 1 2 − ∆k (1 − ∆k+1 ) − (1 − ∆k ) = − 1 + ∆k 2 − ∆k + 2 d˜k+1 =

∆k 2 − ∆k > 0. + 2 2d˜k+1

7.5.2 Picking a “Good” ω As in Section 7.4.2, we would like to minimize α with respect to ω, which, in the case of ε˜ = 12 , can be well approximated from above by ³

¶−1 µ ¶−1 ´µ 1 n−1 1 ˜ ˜ 2dn + 1 ∆n−1 − ≈ 2dn − . 2n−1 ω · 2n−1 dñ

We look for


251

min ω · 2n

ω>n−1

µ

1

−

2n−1

n−1 ω · 2n−1

¶−1

;

that is, min

ω>n−1

or equivalently

ω2 ω−n+1

min (2 ln ω − ln(ω − n + 1)) .

ω>n−1

Setting the gradient to 0, we obtain 2 1 2(ω − n + 1) − ω − = =0 ω ω−n+1 ω(ω − n + 1) which gives us the minimizer ω = 2(n − 1) with the corresponding value of α ≈ (n − 1)22n+1 . This results in di = (n − 1)2n−i+2 , and e hi = O

µ

n22n (2ε)i−1

¶

,

i = 1, . . . , n

i = 1, . . . , n.

7.5.3 An Explicit Formula For a given n, ε, δ, set di = (n − 1)2n−i+2 for i = 1, . . . , n and compute the solution to Ae h = 1 using (7.5). Set ⎢⎛ ⎞ ⎥ ⎢ ⎥ n X ⎢ δ 3 e⎥ ⎣ ⎝ ⎠ h= 1 + max |aij | h⎦ . 3 i j=1 δ From (7.9) it follows that for large n ⎛ ⎞ 0 µ ¶i−1 ⎜1⎟ 1 α −1 ⎜ ⎟ e , hi ≈ B ⎜ . ⎟ ≤ α . d1 2ε ⎝.⎠

i = 1, . . . , n.

1

We Pn choose ε, δ, to minimize the total number of the redundant constraints, i=1 hi . Recalling ε + δ < 1/2, for large n we can write

252


µ ¶i−1 n n n X 3 Xe 3X 1 hi ≈ α hi ≈ δ i=1 δ i=1 2ε i=1 ¡ 1 ¢n 2 1 − 2ε ≈ 3(n − 1)22n+1 1 1 − 2ε 1 − 2ε ¡ 1 ¢n −1 = 3(n − 1)22n+2 2ε 2ε (2ε − 1)2 ¡ 1 ¢n −1 ≤ 3(n − 1)22n+2 2ε . (2ε − 1)2

(7.13)

Pn In fact, we would expect ε to be close to 1/2 in order for i=1 hi to be minimized, so the last inequality also gives us a good approximation, namely ¡ 1 ¢n n X −1 2n+2 2ε hi ≈ 3(n − 1)2 . (7.14) (2ε − 1)2 i=1 Indeed, denoting ζ := 2ε and introducing f (ζ) :=

³ ń 1 ζ

−1

(ζ − 1)2

we can write the last two lines in (7.13) as 3(n − 1)22n+2 ζf (ζ) ≤ 3(n − 1)22n+2 f (ζ). Differentiating ζf (ζ) we get (ζf (ζ))0 =

(n + 1 − ζ n )(ζ + 1) − 2n 0 are positive constants, and |v| denotes the Euclidean norm of v. The criticality condition δP (u) = 0 leads to a coupled nonlinear algebraic system in Rn : ¶ µ 1 2 |Bu| − λ B T Bu = f. (8.48) α 2 Clearly, it is difficult to solve this nonlinear system by direct methods. Also, due to the nonconvexity of P (u), any solution to this nonlinear system satisfies only a necessary condition. The nonconvex function W (v) = 12 α( 12 |v|2 − λ)2 is a so-called double-well energy, which was first studied by van der Waals in fluid mechanics in 1895 (see Rowlinson, 1979). For each given parameter λ > 0, W (v) has two minimizers and one local maximizer (see Figure 8.2a). The global and local minimizers depend on the input f (see Figure 8.2b). This double-well function has extensive applications in mathematical physics. In phase transitions of shape memory alloys, or in the mathematical theory of superconductivity, W (v) is the well-known Landau second-order free energy, and each of its local minimizers represents a possible phase state of the material. In quantum mechanics, if v represents the Higgs’ field strength, then W (v) is the energy. It was discovered in the context of postbuckling analysis of large deformed beam models, that the total potential is also a double-well energy (see Gao, 2000d), and each potential well represents a possible buckled beam state. More examples can be found in a recent review article (Gao, 2003b).

f >0

(a) Graph of W (u) =

1 1 2 ( u 2 2

− λ)2

f 0, and vector f ∈ Rn , the canonical dual function (8.56) has at most three critical points ς¯i (i = 1, 2, 3) satisfying ς¯1 > 0 > ς¯2 ≥ ς¯3 . (8.58) For each of these roots, the vector u ¯i = (B T B)−1 f /¯ ςi ,

for i = 1, 2, 3,

(8.59)

is a critical point of the nonconvex function P (u) in Problem (8.47), and we have P (¯ ui ) = P d (¯ ςi ), ∀i = 1, 2, 3. (8.60) The original version of this theorem was first discovered in a postbifurcation problem of a large deformed beam model in 1997 (Gao, 1997), which shows that there is no duality gap between the nonconvex function P (u) and its canonical dual P d (ς). The dual algebraic equation (8.57) can be solved exactly to obtain all critical points, therefore the vector {¯ ui } defined by (8.59) yields a complete set of solutions to the nonlinear algebraic system (8.48).

8 Canonical Duality Theory

281

τ τ 2 > τc2

0.4

τ 2 = τc2

0.2 τ 2 < τc2

-1

-0.8 -0.6 -0.4 -0.2

0.2

ς

0.4

-0.2

-0.4 Fig. 8.3 Graph of the dual algebraic equation (8.57) and a geometrical proof of the triality theorem.

Let τ 2 = f T (B T B)−1 f . In algebraic geometry, the graph of the algebraic equation τ 2 = 2ς 2 (α−1 ς +λ) is the so-called singular algebraic curve in (ς, τ )space (i.e., the point ς = 0 is on the curve; cf. Silverman and Tate, 1992). From this algebraic curve, we can see that there exists a constant τc such that if τ 2 > τc2 , the dual algebraic equation (8.57) has a unique solution ς > 0. It has three real solutions if and only if τ 2 < τc2 . It is interesting to note that for ς > 0, the total complementary function Ξ(u, ς) is a saddle function and the well-known saddle min-max theory leads to min max Ξ(u, ς) = Ξ(¯ u, ς¯) = max min Ξ(u, ς). (8.61) u

ς>0

ς>0

u

This means that u ¯1 is a global minimizer of P (u) and ς¯1 is a global maximizer on the open domain ς > 0. However, for ς < 0, the total complementary function Ξ(u, ς) is concave in both u and ς < 0; that is, it is a supercritical function. Thus, by the biduality theory, we have that either min max Ξ(u, ς) = Ξ(¯ u, ς¯) = min max Ξ(u, ς) u

ς 0 > ς¯2 > ς¯3 such that u ¯1 is a global minimizer, u ¯2 is a local minimizer, and u ¯3 is a local maximizer of P (u).

8.5.3 Canonical Dual Solutions to Nonconvex Variational Problems Similar to the nonconvex optimization problem (8.47) with the double-well function, let us now consider the following typical nonconvex variational problem, ( ) ¶2 µ Z 1 Z 1 1 02 1 (P) : min P (u) = dx − uf dx , (8.64) α u −λ u∈Uk 2 0 2 0 where f (x) is a given function, λ > 0 is a parameter, and Uk = {u ∈ L2 [0, 1]| u0 ∈ L4 [0, 1], u(0) = 0} is an admissible space. Compared with Problem (8.47), we see that the linear operator B in this case is a differential operator d/dx. This variational problem appears frequently in association with phase transitions in fluids and solids, and in postbuckling analysis of large deformed structures. The criticality condition δP (u) = 0 leads to a nonlinear differential equation in the domain (0, 1) with the natural boundary condition at x = 1; that is,


283

6 ∙T ∙T ∙ T T∙ ∙ ∙∙T ∙ T∙ ∙ ∙

∙∙T ∙ T T -

Fig. 8.5 Zigzag function: Solution to the nonlinear boundary value problem (8.65).

¶¸0 ∙ µ 1 02 0 + f (x) = 0, ∀x ∈ (0, 1), αu u −λ 2 ¶ µ 1 02 0 αu u − λ = 0 at x = 1. 2

(8.65) (8.66)

Due to its nonlinearity, a solution to this boundary value problem is not unique. Particularly, if we let √ f (x) = 0, the equation (8.65) could have three√real roots u0 (x) = {0, ± 2λ}. Thus, any zigzag curve u(x) with slope {0, ± 2λ} solves the boundary value problem, but may not be a global minimizer of the total energy P (u). This problem shows an important fact that in nonconvex analysis the criticality condition is only necessary, but not sufficient for solving variational problems. Traditional direct approaches for solving nonconvex variational problems are very difficult, or impossible. However, by using the canonical dual transformation, this problem can be solved completely. To see this, we introduce a new “strain measure” ξ = Λ(u) = such that the canonical functional Z V (ξ) =

0

1

1 02 u , 2

1 α(ξ − λ)2 dx 2

is convex on Va = {ξ ∈ L2 [0, 1] | ξ(x) ≥ 0 ∀x ∈ (0, 1)}, and the duality relation ς = δV (ξ) = α(ξ − λ) is one-to-one. Thus, its Legendre conjugate can be simply obtained as ½Z 1 ¾ V ∗ (ς) = sta ξς dx − V (ξ) : ξ ∈ Va 0 ¶ Z 1µ 1 −1 2 = α ς + λς dx. 2 0

284

D.Y. Gao, H.D. Sherali

Similar to (8.52), the total complementary function is ¶ Z 1 Z 1µ 1 02 1 −1 2 u ς − α ς − λς dx − uf dx. Ξ(u, ς) = 2 2 0 0

(8.67)

For a given ς 6= 0, the canonical dual functional can be obtained as ¶ Z 1µ 2 τ 1 −1 2 d dx, (8.68) P (ς) = sta{Ξ(u, ς) : u ∈ Uk } = − + λς + α ς 2ς 2 0 where τ (x) is defined by τ =−

Z

x

f (x) dx + c,

(8.69)

0

and the integral constant c depends on the boundary condition. The criticality condition δP d (ς) = 0 leads to the dual equilibrium equation 2ς 2 (α−1 ς + λ) = τ 2 .

(8.70)

This algebraic equation is the same as (8.57), which can be solved analytically as stated below. Theorem 8.7. (Analytical Solutions and Triality Theorem (Gao, 1998a, 2000b)) For any given input function f (x) such that τ (x) is defined by (8.69), the dual algebraic equation (8.70) has at most three real roots ςi (i = 1, 2, 3) satisfying ς¯1 (x) > 0 > ς¯2 (x) ≥ ς¯3 (x). For each ς¯i , the function u ¯i (x) =

Z

x

0

τ dx ς¯i

(8.71)

is a critical point of the variational problem (8.64). Moreover, u ¯1 (x) is a ¯3 (x) is a local maximizer; global minimizer, u ¯2 (x) is a local minimizer, and u that is, P (¯ u1 ) = min max Ξ(u, ς) = max min Ξ(u, ς) = P d (¯ ς1 );

(8.72)

ς2 ); P (¯ u2 ) = min max Ξ(u, ς) = min max Ξ(u, ς) = P d (¯

(8.73)

P (¯ u3 ) = max max Ξ(u, ς) = max max Ξ(u, ς) = P d (¯ ς3 ).

(8.74)

u

u

ς>0

ς∈(¯ ς3 ,0)

u

ς0

u

ς∈(¯ ς3 ,0)

ς 0 if and only if ς > 0. Thus, the total complementary function Ξ(u, ς) given by (8.52) is a saddle function for ς > 0. This leads to the saddle min-max duality (8.61) in the triality theory. Example 8.2. In the nonconvex variational problem (8.64), the quadratic differential operator ξ = Λ(u) = 12 u02 has a physical meaning. In finite deformation theory, if u is considered as the displacement of a deformed body, then ξ can be considered as a Cauchy—Green strain measure (see the following section). The Gâteaux derivative of the quadratic differential operator Λ(u) is Λt (u) = u0 d/dx. For any given u ∈ Ua , using integration by parts, we get hΛt (u)u; ςi =

Z

0

1

u02 ς dx = uu0 ς|x=1 x=0 −

which gives the adjoint operator Λ∗t via ½ 0 uς ∗ Λt (u)ς = 0 [u0 ς] ,

Z

0

1

0

u [u0 ς] dx = hu, Λ∗t (u)ςi,

on x = 1 ∀x ∈ (0, 1).

For any given ς ∈ Va , the Λ-conjugate transformation U Λ (ς) = sta{hΛ(u), ςi − U (u) : u ∈ Uk } = −

Z

1

τ 2 ς −1 dx.

0

The complementary operator in this problem is Λc (u) = Λ(u) − Λt (u)u = − 12 u02 , which leads to the complementary gap function Gc (u, ς) =

Z

0

1

1 02 u ς dx. 2

Clearly, this is positive if ς ≥ 0.

8.6.2 Extremality Conditions: Triality Theory In order to study the extremality conditions of the nonconvex problem, we need to clarify the convexity of the canonical function V (ξ). Without loss of generality, we assume that V : Va → R is convex. Thus, for each u ∈ Ua , the total complementary function Ξ(u, ς) = hΛ(u) ; ςi − V ∗ (ς) − U (u) : Va∗ → R

290


is concave in ς ∈ Va∗ . The convexity of Ξ(·, ς) : Ua → R will depend on the geometrical operator Λ(u) and the function U (u). We furthermore assume that the function Gς (u) = hΛ(u) ; ςi − U (u) : Ua → R is twice Gâteaux differentiable on Ua and let G := {(u, ς) ∈ Ua × Va∗ | δ 2 Gς (u; δu2 ) 6= 0, ∀δu 6= 0}, G + := {(u, ς) ∈ Ua × Va∗ | δ 2 Gς (u; δu2 ) > 0, ∀δu = 6 0}, G − := {(u, ς) ∈ Ua × Va∗ | δ 2 Gς (u; δu2 ) < 0, ∀δu = 6 0}.

(8.93) (8.94) (8.95)

Theorem 8.9. (Triality Theorem) Suppose that (¯ u, ς¯) ∈ G is a critical point of Ξ(u, ς) and Uo × Vo∗ ⊂ Uk × Vk∗ is a neighborhood of (¯ u, ς¯). If (¯ u, ς¯) ∈ G + , then (¯ u, ς¯) is a saddle point of Ξ(u, ς); that is, min max Ξ(u, ς) = Ξ(¯ u, ς¯) = max∗ min Ξ(u, ς).

u∈Uo ς∈Vo∗

ς∈Vo u∈Uo

(8.96)

If (¯ u, ς¯) ∈ G − , then (¯ u, ς¯) is a supercritical point of Ξ(u, ς), and we have that either min max∗ Ξ(u, ς) = Ξ(¯ u, ς¯) = min∗ max Ξ(u, ς) (8.97) u∈Uo ς∈Vo

ς∈Vo u∈Uo

holds, or max max Ξ(u, ς) = Ξ(¯ u, ς¯) = max∗ max Ξ(u, ς).

u∈Uo ς∈Vo∗

ς∈Vo u∈Uo

(8.98)

Proof. By the assumption on the canonical function V (ξ), we know that Ξ(u, ς) is concave on Va∗ . Because Gς (u) is twice Gˆ ateaux differentiable on Ua , the theory of implicit functions tells us that if (¯ u, ς¯) ∈ G, then there exists a unique u ∈ Uo ⊂ Uk such that the dual feasible set Vk∗ is nonempty. If such a point (¯ u, ς¯) ∈ G + , then Gς (u) is convex in u and (¯ u, ς¯) is a saddle point of Ξ on Uo × Vo∗ . The saddle-Lagrangian duality leads to (8.96). If (¯ u, ς¯) ∈ G − , then Gς (u) is locally concave in u and (¯ u, ς¯) is a supercritical point of Ξ(u, ς) on Uo × Vo∗ . In this case the biduality theory leads to (8.97) and (8.98). u t If the geometrical operator Λ(u) is a quadratic function and U (u) is either quadratic or linear, then the second-order Gâteaux derivative δ 2 Gς (u) does not depend on u. In this case, we let ∗ V+ := {ς ∈ Va∗ | δ 2 Gς (u) is positive definite}, ∗ V− := {ς ∈ Va∗ | δ 2 Gς (u) is negative definite}.

(8.99) (8.100)

The following theorem provides extremality criteria for critical points of Ξ(u, ς). Theorem 8.10. (Triduality Theorem (Gao, 1998a, 2000a)) Suppose that Gς (u) = hΛ(u); ςi − U (u) is a quadratic function of u ∈ Ua and (¯ u, ς¯) is a critical point of Ξ(u, ς). ∗ If ς¯ ∈ V+ , then u ¯ is a global minimizer of P (u) on Uk if and only if ς¯ is a ∗ global maximizer of P d (ς) on V+ , and


291

P (¯ u) = min P (u) = max∗ P d (ς) = P d (¯ ς ). u∈Uk

ς∈V+

(8.101)

∗ If ς¯ ∈ V− , then on the neighborhood Uo × Vo∗ ⊂ Ua × Va∗ of (¯ u, ς¯), we have that either P (¯ u) = min P (u) = min∗ P d (ς) = P d (¯ ς) (8.102) u∈Uo

ς∈Vo

holds, or P (¯ u) = max P (u) = max∗ P d (ς) = P d (¯ ς ). u∈Uo

ς∈Vo

(8.103)

∗ provides a This theorem shows that the canonical dual solution ς¯ ∈ V+ global optimality condition for the nonconvex primal problem, whereas the ∗ condition ς¯ ∈ V− provides local extremality conditions. The triality theory was originally discovered in nonconvex mechanics (Gao, 1997, 1999c). Since then, several modified versions have been proposed in nonconvex parametrical variational problems (for quadratic Λ(u) and linear U (u) (Gao, 1998a)), general nonconvex systems (for nonlinear Λ(u) and linear U (u) (Gao, 2000a)), global optimization (for general nonconvex functions of type Φ(u, Λ(u)) (Gao, 2000c), quadratic U (u) (Gao, 2003a,b)), and dissipative Hamiltonian system (for nonconvex/nonsmooth functions of type Φ(u, u,t , Λ(u)) (Gao, 2001c)). In terms of the parametrical function Gς (u) = hΛ(u); ςi−U (u), the current version (Theorems 8.9 and 8.10) can be used for solving general nonconvex problem (8.75) with the canonical function U (u).

8.6.3 Complementary Variational Principles in Finite Deformation Theory In finite deformation theory, the deformation u(x) is a smooth, vector-valued mapping from an open, simply connected, and bounded domain Ω ⊂ Rn into a deformed domain2 ω ⊂ Rm . Let Γ = ∂Ω = Γu ∪ Γt be the boundary of ¯ is prescribed, whereas Ω such that on Γu , the boundary condition u(x) = u on the remaining boundary Γt , the surface traction (external force) ¯t(x) is applied. Similar to the nonconvex optimization problem (8.48), the primal problem is to minimize the total potential energy functional: ½ ¾ Z Z ¯ ¯ on Γu , min P (u) = [W (∇u) − u · f ] dΩ − u · t dΓ : u = u Ω

Γt

(8.104) where the stored energy W (F) is a Gˆ ateaux differentiable function of F = ∇u, and f (x) is a given force field. Because the deformation gradient F = ∇u ∈

2 If m = n + 1, then the deformation u(x) represents a hypersurface in m-dimensional space. Applications of the canonical duality theory in differential geometry were discussed in Gao and Yang (1995).

292


Rn×m is a so-called two-point tensor, which is no longer a strain measure in finite deformation theory, the stored energy W (F) is usually nonconvex. Particularly, for St. Venant—Kirchhoff material (see Gao, 2000a), we have ¸ ∙ ¸ ∙ 1 1 T 1 T W (²) = (8.105) (F F − I) : D : (F F − I) , 2 2 2 where I is an identity tensor in Rn×n . Due to nonconvexity, the duality relation τ = δW (F) is not one-to-one. Although the two-point tensor τ ∈ Rm×n is called the first Piola—Kirchhoff stress, according to Hill’s constitutive theory, (F, τ ) is not considered as a work-conjugate (canonical) strain—stress pair (see Gao, 2000a). The Fenchel—Rockafellar type dual variational problem is ½ ¾ Z Z ¯ · τ · n dΓ − max P (τ ) = W (τ ) dΩ (8.106) u Γu

s.t. −∇ · τ

T

= f in Ω, n · τ

Ω

T

= ¯t on Γt .

(8.107)

In the case where the stored energy W (F) is convex, then W (τ ) = W ∗ (τ ) which is called the complementary energy in elasticity. In this case, the functional Z Z c ∗ ¯ · τ · n dΓ Π (τ ) = W (τ ) dΩ − u Ω

Γu

is the well-known Levinson—Zubov complementary energy. As discussed before, if the stored energy W (F) is nonconvex, the Legendre conjugate W ∗ is not uniquely defined. It turns out that the Levinson—Zubov complementary variational principle can be used only for solving convex problems (see Gao, 1992). Although the Fenchel conjugate W (τ ) can be uniquely defined, the Fenchel—Young inequality W (F) + W (τ ) ≥ hF; τ i leads to a duality gap between the minimal potential variational problem (8.104) and its Fenchel— Rockafellar dual (see Gao, 1992); that is, in general, min P (u) ≥ max P (τ ).

(8.108)

By the fact that the criticality condition δP (τ ) = 0 is not equivalent to the primal variational problem and the weak duality is not appreciated in the field of continuum mechanics, the existence of a perfect (i.e., without a duality gap), pure (i.e., involving only stress tensor as variational argument) complementary variational principle in finite elasticity has been argued among well-known scientists for more than three decades (see Hellinger, 1914, Hill, 1978, Koiter, 1973, 1976, Lee and Shield, 1980a,b, Levinson, 1965, Ogden, 1975, 1977, Zubov, 1970). This problem was finally solved by the canonical dual transformation and triality theory in Gao (1992, 1999c).


293

Similar to the quadratic operator Λ(u) = 12 |Bu|2 (see equation (8.51)) chosen for the nonconvex optimization problem (8.48), we let E = Λ(u) =

1 [(∇u)T (∇u) − I], 2

(8.109)

which is a symmetrical tensor field in Rn×n . In finite deformation theory, E is the well-known Green—St. Venant strain tensor. Thus, in terms of E, the stored energy for St. Venant—Kirchhoff material can be written in the canonical form W (∇u) = V (Λ(∇u)), and V (E) =

1 E:D:E 2

is a (quadratic) convex function of the symmetrical tensor E ∈ Rn×n . The canonical dual variable E∗ = δV (E) = D · E is called the second Piola— Kirchhoff stress tensor, denoted as T. The Legendre conjugate V ∗ (T) =

1 T : D−1 : T 2

(8.110)

¯ on Γu } is also a quadratic function. Let Ua = {u ∈ W 1,p (Ω; R3 ) | u = u (where W 1,p is a standard Sobolev space with p ∈ (1, ∞)) and Va∗ = C(Ω; Rn×n ). Replacing W (∇u) by its canonical dual transformation V (Λ(u)) = E(u) : T − V ∗ (T), the generalized complementary energy Ξ : Ua × Va∗ → R has the following format, Z Z Ξ(u, T) = [E(u) : T − V ∗ (T) − u · f ] dΩ − u · ¯t dΓ, (8.111) Ω

Γt

which is the well-known Hellinger—Reissner generalized complementary energy in continuum mechanics. Furthermore, if we replace V ∗ (T) by its bi-Legendre transformation E : T − V (E), then Ξ(u, T) can be written as Z Z Ξhw (u, T, E) = [Λ(∇u) − E) : T + V (E) − u · f ] dΩ − u · ¯t dΓ. (8.112) Ω

Γt

This is the well-known Hu—Washizu generalized potential energy in nonlinear elasticity. The Hu—Washizu variational principle has important applications in computational analysis of thin-walled structures, where the geometrical equation E = Λ(u) is usually proposed by certain geometrical hypothesis. ¯ in the Because Λ(u) is a quadratic operator, its Gâteaux differential at u direction u is δΛ(¯ u; u) = Λt (¯ u)u = (∇¯ u)T (∇u) and 1 Λc (u) = Λ(u) − Λt (u)u = − [(∇u)T (∇u) + I]. 2

294


By using the Gauss—Green theorem, the balance operator Λ∗t (u) can be defined as ½ −∇ · [(∇u)T · T]T in Ω, Λ∗t (u)T = n · [(∇u)T · T]T on Γ. The complementary gap function in this problem is a quadratic functional: Z 1 Gc (u, T) = h−Λc (u); Ti = (8.113) tr[(∇u)T · T · (∇u) + T] dΩ. Ω 2 Thus, the complementary variational problem is to find critical (stationary) ¯ such that points (¯ u, T) ½Z ¾ Z 1 c T ∗ ¯ P (¯ u, T) = sta V (T) dΩ (8.114) tr[(∇u) · T · (∇u) + T] dΩ + Ω 2 Ω s.t. −∇ · [(∇u)T · T]T = f in Ω, n · [(∇u)T · T]T = ¯t on Γt . The following result is due to Gao and Strang in 1989 (1989a). Theorem 8.11. (Complementary—Dual Variational Principle (Gao ¯ is a critical point of the complementary and Strang, 1989a)) If (¯ u, T) ¯ is a critical point of the total potential variational problem (8.114), then u energy P (u) defined by (8.104), and ¯ = 0. P (¯ u) + P c (¯ u, T) Moreover, if the complementary gap function ¯ ≥ 0, Gc (u, T)

∀u ∈ Ua ,

(8.115)

¯ is a global minimizer of P (u) and then u ¯ P (¯ u) = min P (u) = max min Ξ(u, T) = −P c (¯ u, T), u

T

u

(8.116)

subject to T(x) being positive definite for all x ∈ Ω. This theorem shows that the positivity of the complementary gap function Gc (u, T) provides a sufficient condition for a global minimizer, and the equalities (8.11) and (8.116) indicate that there is no duality gap between the total potential P (u) and its complementary energy P c (u, T). The physical significance is also clear: a finite deformed material is stable if the second Piola—Kirchhoff stress tensor T(x) is positive definite everywhere in the domain Ω. The linear operator B = ∇ in this nonconvex variational problem is a partial differential operator, therefore it is difficult to find its inverse. It took more than ten years before the canonical dual problem was finally formulated in Gao (1999a,c). To see this, let us assume that for a given force vector field ¯t on the boundary Γt , the first Piola—Kirchhoff stress tensor τ (x) can be defined by solving the following boundary value problem,


295

−∇ · τ T (x) = f in Ω,

n · τ T = ¯t on Γt .

Then the canonical dual functional P d (T) can be formulated as Z Z 1 d −1 T P (T) = − V ∗ (T) dΩ. tr(τ · T · τ + T) dΩ − Ω 2 Ω

(8.117)

(8.118)

The criticality condition δP d (T) = 0 gives the canonical dual equation T · (2 δV ∗ (T) + I) · T = τ T · τ .

(8.119)

For St. Venant—Kirchhoff material, V ∗ (T) = 12 T : D−1 : T is a quadratic function and its Gâteaux derivative δV ∗ (T) = D−1 · T is linear. In this case, the canonical dual equation (8.119) is a cubic equation, which is similar to the dual algebraic equations (8.57) and (8.70). Theorem 8.12. (Pure Complementary Energy Principle (Gao, 1999a,c)) Suppose that for a given force field ¯t(x) on Γt , the first Piola— ¯ of the Kirchhoff stress field τ (x) is defined by (8.117). Then each solution T canonical dual equation (8.119) is a critical point of P d , the vector defined by the line integral Z ¯ −1 dx ¯ = τ ·T u (8.120)

is a critical point of P (u), and

¯ P (¯ u) = P d (T). This theorem presents an analytic solution to the nonconvex potential variational problem (8.104). In the finite deformation theory of elasticity, this pure complementary variational principle is also known as the Gao principle (Li and Gupta, 2006), which holds also for the general canonical energy function V (E). Similar to Theorem 8.9, the extremality of the critical points can be identified by the complementary gap function. Applications of this pure complementary variational principle for solving nonconvex/nonsmooth boundary value problems are illustrated in Gao (1999c, 2000a) and Gao and Ogden (2008a,b).

8.7 Applications to Semilinear Nonconvex Systems The canonical dual transformation and the associated triality theory can be used to solve many difficult problems in engineering and science. In this section, we present applications for solving the following nonconvex minimization problem

296


(P) :

1 min{P (u) = W (u) + hu, Aui − hu, f i : u ∈ Uk }, 2

(8.121)

where W (u) : Uk → R is a nonconvex function, and A : Ua ⊂ U → Ua∗ is a linear operator. If W (u) is Gâteaux differentiable, the criticality condition δP (u) = 0 leads to a nonlinear Euler equation Au + δW (u) = f.

(8.122)

The abstract form (8.122) of the primal problem (P) covers many situations. In nonconvex mechanics (cf. Gao, Ogden, and Stavroulakis, 2001, Gao, 2003b), where U is an infinite-dimensional function space, the state variable u(x) is a field function, and A : U → U ∗ is usually a partial differential operator. In this case, the governing equation (8.122) is a so-called semilinear equation. For example, in the Landau—Ginzburg theory of superconductivity, A = ∆ is the Laplacian over a given space domain Ω ⊂ Rn and W (u) =

Z

Ω

1 α 2

µ

¶2 1 2 dΩ u −λ 2

(8.123)

is the Landau double-well potential, in which α, λ > 0 are material constants. Then the governing equation (8.122) leads to the well-known Landau— Ginzburg equation 1 ∆u + αu( u2 − λ) = f. 2 This semilinear differential equation plays an important role in material science and physics including: ferroelectricity, ferromagnetism, and superconductivity. In a more complicated case where A = ∆ + curl curl, we have 1 ∆u + curl curl u + αu( u2 − λ) = f, 2 which is the so-called Cahn—Hilliard equation in liquid crystal theory. Due to the nonconvexity of the double-well function W (u), any solution of the semilinear differential equation (8.122) is only a critical point of the total potential P (u). Traditional direct analysis and related numerical methods for finding the global minimizer of the nonconvex variational problem have proven unsuccessful to date. In dynamical systems, if A = −∂,tt + ∆ is a wave operator over a given space—time domain Ω ⊂ Rn × R, then (8.122) is the well-known nonlinear Schrödinger equation 1 −u,tt + ∆u + αu( u2 − λ) = f. 2 This equation appears in many branches of physics. It provides one of the simplest models of the unified field theory. It can also be found in the theory


297

(a) u(t)

(b) Trajectory in phase space u−p

4

2

3 1

2 1

0 0 −1

−1

−2 −3

0

10

20

30

40

−2 −4

(a) u(t)

−2

0

2

4

(b) Trajectory in phase space u−p

4

2

3 1

2 1

0 0 −1

−1

−2 −3

0

10

20

30

40

−2 −4

−2

0

2

4

Fig. 8.7 Numerical results by ode23 (top) and ode15s (bottom) solvers in MATLAB.

of dislocations in metals, in the theory of Josephson junctions, as well as in interpreting certain biological processes such as DNA dynamics. In the most simple case where u depends only on time, the nonlinear Schrödinger equation reduces to the well-known Duffing equation 1 u,tt = αu( u2 − λ) − f. 2 Even for this one-dimensional ordinary differential equation, an analytic solution is still very difficult to obtain. It is known that this equation is extremely sensitive to the initial conditions and the input (driving force) f (t). Figure 8.7 displays clearly that for the same given data, two Runge—Kutta solvers in MATLAB produce very different vibration modes and “trajectories” in the phase space u—p (p = u,t ). Mathematically speaking, due to the nonconvexity of the function W (u), very small perturbations of the system’s initial conditions and parameters may lead the system to different local minima with significantly different performance characteristics, that is, the so-called chaotic phenomena. Numerical results vary with the methods used. This is one of the main reasons why traditional perturbation analysis and direct approaches cannot successfully be applied to nonconvex systems (Gao, 2003b).

298


Numerical discretization of the nonconvex variational problem (P) in mathematical physics usually leads to a nonconvex optimization problem in finite-dimensional space U = Rn , where the field variable u is simply a vector x ∈ U, the bilinear form hx, x∗ i = xT x∗ = x · x∗ is the dot-product of two vectors, and the operator A : Rn → U ∗ = Rn is a symmetrical matrix. In d.c. (difference of convex functions) programming and discrete dynamical systems, the operator A = AT ∈ Rn×n is usually indefinite. The problem (8.121) is then one of global minimization in Rn . In this section, we discuss the canonical dual transformation method for solving this type of problem.

8.7.1 Unconstrained Nonconvex Optimization Problem with Double-Well Energy First, let us consider an unconstrained global optimization problem in finitedimensional space U = Rn , where A = AT ∈ Rn×n is a matrix, and W (x) is a double-well function of the type W (x) = 12 ( 12 |x|2 − λ)2 . Then the primal problem is ( ) ¶2 µ 1 1 2 1 T T n min P (x) = . |x| − λ + x Ax − x f : ∀x ∈ Uk = R 2 2 2 (8.124) The necessary condition δP (x) = 0 leads to a coupled nonlinear algebraic system µ ¶ 1 2 Ax + (8.125) |x| − λ x = f. 2 Clearly, a direct method for solving this nonlinear equation with n unknown is elusive. By choosing the quadratic operator ξ = 12 |x|2 , the canonical function V (ξ) = 12 (ξ − λ)2 is a quadratic function. By the fact that 12 |x|2 = ξ ≥ 0, ∀x ∈ Rn , the range of the quadratic mapping Λ(x) is Va = {ξ ∈ R| ξ ≥ 0}. Thus, on Va , the canonical duality relation ς = δV (ξ) = ξ − λ is one-to-one and the range of the canonical dual mapping δV : Va → V ∗ ⊂ R is Va∗ = {ς ∈ R| ς ≥ −λ}. It turns out that (ξ, ς) is a canonical pair on Va × Va∗ and the Legendre conjugate V ∗ is also a quadratic function: V ∗ (ς) = sta{ξς − V (ξ) : ξ ∈ Va } = For a given ς ∈ Va∗ , the Λ-conjugate transformation

1 2 ς + λς. 2


299

U Λ (ς) = sta

½

1 2 1 x ς − xT Ax + xT f : x ∈ Rn 2 2

1 −1 = − f T (A + ςI) f 2

¾

is well defined on the canonical dual feasible space Vk∗ , given by Vk∗ = {ς ∈ R| det(A + ςI) 6= 0, ς ≥ −λ}.

(8.126)

Thus, the canonical dual problem can be proposed as the following (Gao, 2003a): ½ ¾ 1 1 (P d ) : max P d (ς) = − f T (A + ςI)−1 f − ς 2 − λς : ς ∈ Vk∗ . 2 2 (8.127) This is a nonlinear programming problem with only one variable! The criticality condition of this dual problem leads to the dual algebraic equation ς +λ=

1 T f (A + ςI)−2 f. 2

(8.128)

For any given A ∈ Rn×n and f ∈ Rn , this equation can be solved by Mathematica. Extremality conditions of these dual solutions can be identified by the following theorem (see Gao, 2003a). Theorem 8.13. (Gao, 2003a) If the matrix A has r distinct nonzero eigenvalues such that a1 < a2 < · · · < ar , then the canonical dual algebraic equation (8.128) has at most 2r + 1 roots ς1 > ς2 ≥ ς3 ≥ · · · ≥ ς2r+1 . For each ςi , the vector xi = (A + ςi I)−1 f,

∀i = 1, 2, . . . , 2r + 1,

(8.129)

is a solution to the semilinear algebraic equation (8.125) and P (xi ) = P d (ςi ),

∀i = 1, . . . , 2r + 1.

(8.130)

Particularly, the canonical dual problem has at most one global maximizer ς1 > −a1 in the open interval (−a1 , +∞), and x1 is a global minimizer of P (x) over Uk ; that is, P (x1 ) = min P (x) = max P d (ς) = P d (ς1 ). x∈Uk

ς>−a1

(8.131)

300


2 1 3 2 1 0 -1

0 2 1

-1

0 -1 -1

0

-2

-2

1

-2

-1

0

1

2

Fig. 8.8 Graph of the primal function P (x1 , x2 ) and its contours.

2 1.5 1 0.5 -1.5

-1

-0.5 -0.5

0.5

1

-1 -1.5 -2 Fig. 8.9 Graph of the dual function P d (ς).

Moreover, in each open interval (−ai+1 , −ai ), the canonical dual equation (8.128) has at most two real roots −ai+1 < ς2i+1 < ς2i < −ai , ∀i = 1, . . . , 2r+ 1, ς2i is a local minimizer of P d , and ς2i+1 is a local maximizer of P d (ς). As an example in two-dimensional space, which is illustrated in Figure 8.8, we simply choose A = {aij } with a11 = 0.6, a22 = −0.5, a12 = a21 = 0, and f = {0.2, −0.1}. For a given parameter λ = 1.5, and α = 1.0, the graph of P (x) is a nonconvex surface (see Figure 8.8a) with four potential wells and one local maximizer. The graph of the canonical dual function P d (ς) is shown in Fig. 8.9. The dual canonical dual algebraic equation (8.128) has a total of five real roots: ς¯5 = −1.47 < ς¯4 = −0.77 < ς¯3 = −0.46 < ς¯2 = 0.45 < ς¯1 = 0.55, and we have


301 5

2.5

-3

-2

-1

1

2

3

-2.5

-5

-7.5

-10

-12.5

Fig. 8.10 Graph of the dual function P d (ς) for a four-dimensional problem.

P d (¯ ς5 ) = 1.15 > P d (¯ ς4 ) = 0.98 > P d (¯ ς3 ) = 0.44 > P d (¯ ς2 ) = −0.70 > P d (¯ ς1 ). ¯ 1 = (A + ς¯1 I)−1 f = {0.17, −2.02} is By the triality theory, we know that x a global minimizer of P (¯ x); and accordingly, P (¯ x1 ) = P d (¯ ς1 ) = −1.1; and ¯ 5 = {−0.23, 0.05} and x ¯ 3 = {1.44, 0.10} are local maximizers, whereas that x ¯ 4 = {−1.21, 0.08} and x ¯ 2 = {0.19, 1.96} are local minimizers. x The graph of P d (ς) for a four-dimensional problem is shown in Figure 8.10. It can be easily seen that P d (ς) is strictly concave for ς > −a1 . Within each interval −ai−1 < ς < −ai , ∀i = 1, 2, . . . , r, the dual function P d (ς) has at most one local minimum and one local maximum. These local extrema can be identified by the triality theory (Gao, 2003a). The nonconvex function W (x) in (8.121) could be in many other forms, for example, µ ¶ 1 2 W (x) = exp |Bx| − λ , 2 where B ∈ Rm×n is a given matrix and λ > 0 is a constant. In this case, the primal problem (P) is a quadratic-exponential minimization problem ½ µ ¶ ¾ 1 1 T 2 T n min P (x) = exp |Bx| − λ + x Ax − x f : x ∈ R . 2 2 By letting ξ = Λ(x) = 12 |Bx|2 − λ, the canonical function V (ξ) = exp(ξ) is convex and its Legendre conjugate is V ∗ (ς) = ς(ln ς − 1). The canonical dual problem was formulated in Gao and Ruan (2007): ½ ¾ 1 ∗ (P d ) : max P d (ς) = − f T [G(ς)]−1 f − (ς log ς − ς) − λς : ς ∈ V+ , 2 where G(ς) = A + ςB T B and the dual feasible space is defined by

302

D.Y. Gao, H.D. Sherali ∗ V+ = {ς ∈ R | ς > 0, G(ς) is positive definite}.

Detailed study of this case was given in Gao and Ruan (2007).

8.7.2 Constrained Quadratic Minimization over a Sphere If the function W (x) in problem (8.121) is an indicator of a constraint set Uk ⊂ Rn , that is, ½ 0 if x ∈ Uk , W (x) = +∞ otherwise, then the general problem (8.121) becomes a constrained nonconvex quadratic optimization problem, denoted as (Pq ) :

min{P (x) =

1 hx, Axi − hx, f i : x ∈ Uk }. 2

(8.132)

General constrained global optimization problems are discussed in the next section. Here, we consider the following quadratic minimization problem with a nonlinear constraint (Pq ) :

min P (x) =

1 T x Ax − f T x 2

(8.133)

s.t. |x| ≤ r, where A = AT ∈ Rn×n is a symmetric matrix, f ∈ Rn is a given vector, and r > 0 is a constant. The feasible space Uk = {x ∈ Rn | |x| ≤ r} is a hypersphere in Rn . This problem often arises as a subproblem in general optimization algorithms (cf. Powell, 2002). Often, in the model trust region methods, the objective function in nonlinear programming is approximated locally by a quadratic function. In such cases, the approximation is restricted to a small region around the current iterate. These methods therefore require the solution of quadratic programming problems over spheres. To solve this constrained nonconvex minimization by using a traditional Lagrange multiplier method, we have L(x, λ) =

1 T x Ax − f T x + λ(|x| − r). 2

(8.134)

For a given λ ≥ 0, the traditional dual function can be defined via the Fenchel—Moreau—Rockafellar duality theory: P ∗ (λ) = min{L(x, λ) : x ∈ Rn },

(8.135)


303

which is a concave function of λ. However, due to the nonconvexity of P (x), we have only the weak duality relationship min P (x) ≥ max P ∗ (λ).

|x|≤r

λ≥0

The duality gap θ given by the slack in the above inequality is typically nonzero indicating that the dual solution does not solve the primal problem. On the other hand, the KKT condition leads to a coupled nonlinear algebraic system Ax + λ|x|−1 x = f, λ ≥ 0, |x| ≤ r, λ(|x| − r) = 0. As indicated by Floudas and Visweswaran (1995), due to the presence of the nonlinear sphere constraint, the solution of (Pq ) is likely to be irrational, which implies that it is not possible to exactly compute the solution. Therefore, many polynomial time algorithms have been suggested to compute the approximate solution to this problem (see Ye, 1992). However, by the canonical dual transformation this problem has been solved completely in Gao (2004b). First, we need to reformulate the constraint |x| ≤ r in the canonical form ξ = Λ(x) =

1 2 |x| . 2

Let λ = 12 r2 , then the canonical function V (Λ(x)) can be defined as ½ 0 if ξ ≤ λ, V (ξ) = +∞ otherwise, whose effective domain is Va = {ξ ∈ R | ξ ≤ λ}. Letting U (x) = xT f − 12 xT Ax, the primal problem (Pq ) can be reformulated in the following canonical form, min{Π(x) = V (Λ(x)) − U (x) : x ∈ Rn }. By the Fenchel transformation, the conjugate of V (ξ) is ½ λς if ς ≥ 0, V (ς) = max{ξς − V (ξ)} = +∞ otherwise, ξ∈Va

(8.136)

(8.137)

whose effective domain is Va∗ = {ς ∈ R| ς ≥ 0}. The dual feasible space Vk∗ in this problem is Vk∗ = {ς ∈ R | ς ≥ 0, det(A + ςI) 6= 0}. Thus, for a given ς ∈ Va∗ , the Λ-conjugate of U can be formulated as

304

D.Y. Gao, H.D. Sherali Λ

U (ς) = sta

½

1 2 1 |x| ς + xT Ax − xT f : x ∈ Rn 2 2

1 = − f T (A + ςI)−1 f, 2

¾

and the problem (P d ), which is perfectly dual to (Pq ), is given by ½ ¾ 1 T d d −1 ∗ (Pq ) : max P (ς) = − f (A + ςI) f − λς : ς ∈ Vk . 2

(8.138)

The criticality condition δP d (¯ ς ) = 0 leads to a nonlinear algebraic equation 1 T f (A + ς¯I)−2 f = λ. 2

(8.139)

Similar to (8.128), this equation can also be solved easily by using Mathematica. Each root ς¯i is a critical point of P d (ς). The following theorem presents a complete set of solutions for this dual problem. Theorem 8.14. (Complete Solution to (Pq ) (Gao, 2004b)) Suppose that the symmetric matrix A has p ≤ n distinct eigenvalues, and id ≤ p of them are negative such that a1 < a2 < · · · < aid < 0 ≤ aid +1 < · · · < ap . Then for a given vector f ∈ Rn , the canonical dual problem (Pqd ) has at most 2id +1 critical points ς¯i , i = 1, . . . , 2id +1, satisfying the following distribution law, ς¯1 > −a1 > ς¯2 ≥ ς¯3 > −a2 > · · · > −aid > ς¯2id ≥ ς¯2id +1 > 0.

(8.140)

For each ς¯i ≥ 0, i = 1, . . . , 2id + 1, the vector defined by ¯ i = (A + ς¯i I)−1 f x

(8.141)

is a KKT point of the problem (Pq ) and P (¯ xi ) = P d (¯ ςi ),

i = 1, 2, . . . , 2id + 1.

(8.142)

Moreover, if id > 0, then the problem (Pq ) has at most 2id + 1 critical points on the boundary of the sphere; that is, 1 |¯ xi |2 = λ, 2

i = 1, . . . , 2id + 1.

(8.143)

Because A = AT , there exists an orthogonal matrix RT = R−1 such that A = RT DR, where D = (ai δij ) is a diagonal matrix. For the given vector f ∈ Rn , let g = Rf = (gi ), and define


305 5

ψ=λ

4 3 2 1

-4

-2

2

4

-1

Fig. 8.11 Graph of ψ(ς).

ψ(ς) =

1 T f (A + ςI)−2 f 2 p

=

1X 2 g (ai + ς)−2 . 2 i=1 i

(8.144)

Clearly, this real-valued function ψ(ς) is strictly convex within each interval −ai+1 < ς < −ai , as well as over the intervals −∞ < ς < −ap and −a1 < ς < ∞ (see Figure 8.11). Thus, for a given parameter λ > 0, the algebraic equation p 1X 2 ψ(ς) = g (ai + ς)−2 = λ (8.145) 2 i=1 i

has at most 2p solutions {¯ ςi } satisfying −aj+1 < ς¯2j+1 ≤ ς¯2j < −aj for j = 1, . . . , p − 1, and ς¯1 > −a1 , ς¯2p < −ap . Because A has only id negative eigenvalues, the equality ψ(ς) = λ has at most 2id + 1 strictly positive roots {¯ ςi } > 0, i = 1, . . . , 2id + 1. By the complementarity condition ς¯i ( 12 |¯ xi |2 − λ) = 0, we know that the primal problem (Pq ) has at most 2id + 1 KKT ¯ i on the sphere 12 |¯ points x xi |2 = λ. If aid +1 > 0, the equality ψ(ς) = λ may have at most 2id strictly positive roots. By using the triality theory, the extremality conditions of the critical points of the problem (Pq ) can be identified by the following result. Theorem 8.15. (Global and Local Extrema (Gao, 2004b)) Suppose that a1 is the smallest eigenvalue of A. Then the dual problem (P d q ) given ¯ 1 is in (8.138) has a unique solution ς¯1 over the domain ς > −a1 ≥ 0, and x a global minimizer of the problem (Pq ); that is, P (¯ x1 ) = min P (x) = max P d (ς) = P d (¯ ς1 ). x∈Uk

ς>−a1

(8.146)

306


2

1.5

1

0.5

0

-0.5

-1

-1.5

-0.75

-0.5

-0.25

0

−a2i+1

0.25

−a2i

0.5

0.75

1

− a1

Fig. 8.12 Graph of P d (ς).

If in each interval (−ai+1 , −ai ), i = 1, . . . , id , the dual algebraic equation (8.139) has two roots −ai+1 < ς¯2i+1 < ς¯2i < −ai , then ς¯2i is a local minimizer of P d (ς), and ς¯2i+1 is a local maximizer of P d (ς) over the interval (−ai+1 , −ai ). Proof. Because for any given ς > −a1 , the matrix A + ςI is positive definite, that is, the total complementary function Ξ(x, ς) is a saddle function, the saddle minmax theorem leads to (8.146). The remaining statements in Theorem 8.15 can be proved by the graph of P d (ς) (see Figure 8.12). t u It is interesting to note that on the effective domain Va∗ , the Fenchel—Young equality V (ξ) = hξ; ςi − V ∗ (ς) = (ξ − λ)ς holds true. Thus, on Ua × Va∗ , the total complementary function Ξ(x, ς) = hΛ(x); ςi − V ∗ (ς) − U (x) =ς

µ

¶ 1 1 2 |x| − λ + xT Ax − xT f 2 2

(8.147)

can be viewed as the traditional Lagrangian of the quadratic minimization problem with the reformulated (canonical) quadratic constraint 12 |x|2 ≤ λ, which is also called extended Lagrangian (see Gao, 2000a). This example exhibits a connection between the nonlinear Lagrange multiplier method and the canonical dual transformation. Based on this observation, the traditional Lagrange multiplier method can be generalized to solve constrained global optimization problems.


307

8.8 General Constrained Global Optimization Problems In this section, we present an important application of the canonical duality theory to the following general constrained nonlinear programming problem min {−U (x) : x ∈ Uk },

(8.148)

where U (x) is a Gˆ ateaux differentiable function, either a linear or canonical function, defined on an open convex set Ua ⊂ Rn , and the feasible space Uk is a convex subset of Ua defined by Uk = {x ∈ Ua ⊂ Rn | gi (x) ≤ 0, i = 1, . . . , p}, in which gi (x) : Ua → R are convex functions. We show the connection between the canonical dual transformation and nonlinear Lagrange multiplier methods and how to use the triality theory to identify global and local optima.

8.8.1 Canonical Form and Total Complementary Function First, we need to put this problem in the framework of the canonical systems. Let the geometrical operator ξ = Λ(x) = {gi (x)} : Ua → Va ⊂ Rp be a vector-valued function. The generalized canonical function ½ 0 if ξ ≤ 0 V (ξ) = ∞ otherwise is an indicator of the convex cone Va = {ξ ∈ Rp | ξ ≤ 0}. Thus, the canonical form of the constrained problem (8.148) is min{Π(x) = V (Λ(x)) − U (x) : x ∈ Ua }. By the Fenchel transformation, the conjugate of V (ξ) is an indicator of the dual cone Va∗ = {ς ∈ Rp | ς ≥ 0}; that is, ½ 0 if ς ≥ 0 V (ς) = max{hξ; ςi − V (ξ) : ξ ∈ Rp } = ∞ otherwise. By the theory of convex analysis we have ς ∈ ∂ − V (ξ) ⇔ ξ ∈ ∂ − V (ς) ⇔ hξ ; ςi = V (ξ) + V (ς);

(8.149)

that is, (ξ, ς) is a generalized canonical pair on Ua × Va∗ (Gao, 2000c). Thus, the extended Lagrangian Ξ(x, ς) = hΛ(x); ςi − V (ς) − U (x) in this problem has a very simple form:

308


Ξ(x, ς) = −U (x) +

p X

ςi gi (x).

(8.150)

i=1

We can see here that the canonical dual variable ς ≥ 0 ∈ Rp is nothing but a Lagrange multiplier for the constraints Λ(x) = {gi (x)} ≤ 0. Let I(¯ x) := {i ∈ {1, . . . , p}| gi (¯ x) = 0} ¯ . By the theory of global optibe the index set of the active constraints at x ¯ is a local minimizer such mization (cf. Horst et al., 2000) we know that if x that ∇gi (¯ x), i ∈ I(¯ x), are linearly independent, then the KKT conditions hold: gi (¯ x) ≤ 0,

ς¯i ≥ 0,

ς¯i gi (¯ x) = 0, i = 1, . . . , p, p X ∇U (¯ x) = ς¯i ∇gi (¯ x).

(8.151) (8.152)

i=1

Any point (¯ x, ς¯) that satisfies (8.151)—(8.152) is called a KKT stationary point of the problem (8.148). However, the KKT conditions (8.151)—(8.152) are only necessary for the minimization problem (8.148). They are sufficient ¯ provided that, for example, the funcfor a constrained global minimum at x tions P (x) = −U (x) and gi (x), i = 1, . . . , p, are convex. In constrained global optimization problems, the primal problems may possess many local minimizers due to the nonconvexity of the objective function and constraints. Therefore, sufficient optimality conditions play a key role in developing global algorithms. Here we show that the triality theory can provide such sufficient conditions. The complementary function V (ς) = 0, ∀ς ∈ Va∗ , therefore in this constrained optimization problem we have Gς (x) = Ξ(x, ς) = −U (x) + ς T Λ(x).

(8.153)

For a fixed ς ∈ Va∗ , if the parametric function Gς : Ua → R is twice Gâteaux differentiable, the space G can be written as µ 2 ¶ ¾ ½ ∂ Gς (x) ∗ 6= 0 . G = (x, ς) ∈ Ua × Va | det ∂xi ∂xj Clearly for any given (x, ς) ∈ G, the dual feasible space Vk∗ , ( ) p X ∗ ∗ ∗ ςi ∇gi (x) = ∇U (x), ∀x ∈ Ua Vk = ς ∈ Va | Λt (x)ς = i=1

is nonempty and the Λ-conjugate transformation

(8.154)


309

U Λ (ς) = sta {hΛ(x); ςi − U (x) : ∀x ∈ Ua } can be well formulated on Vk∗ . Thus, the canonical dual problem can be proposed as the following, max{P d (ς) = −U Λ (ς) : ς ∈ Vk∗ }.

(8.155)

In the following, we illustrate the foregoing results using some examples.

8.8.2 Quadratic Minimization with Quadratic Constraints Let U (x) = xT f − 12 xT Ax and g(x) = 12 xT Cx − λ be quadratic functions, where A and C are two symmetrical matrices in Rn×n , f ∈ Rn is a given vector, and λ ∈ R is a given constant. Thus the primal problem is: ½ ¾ 1 1 T min P (x) = xT Ax − f T x : (8.156) x Cx ≤ λ, x ∈ Rn . 2 2 Because we have only one constraint g(x) = 12 xT Cx − λ, the extended Lagrangian is simply Ξ(x, ς) =

1 T x (A + ςC)x − f T x − ςλ. 2

(8.157)

On the dual feasible space Vk∗ = {ς ∈ R | ς ≥ 0, det(A + ςC) 6= 0}, and the canonical dual problem (8.155) can be formulated as (see Gao, 2005a): ½ ¾ 1 T d −1 ∗ max P (ς) = − f (A + ςC) f − λς : ς ∈ Vk . (8.158) 2 Because in this problem both Λ(x) = ( 12 xT Cx − λ) and U (x) = − 12 xT Ax + f T x are quadratic functions, δ 2 Gς = (A + ςC). The following result was obtained recently. Theorem 8.16. (Gao, 2005a) Suppose that the matrix C is positive definite, and ς¯ ∈ Va∗ is a critical point of P d (ς). If A + ς¯C is positive definite, the vector ¯ = (A + ς¯C)−1 f x is a global minimizer of the primal problem (8.156). However, if A + ς¯C is ¯ = (A + ς¯C)−1 f is a local minimizer of the negative definite, the vector x primal problem (8.156).

310

D.Y. Gao, H.D. Sherali 3 2 1

20 0

10 2

0 -10

-1 0 -2

-2 0 -2 -3 -3

2

-2

-1

0

1

2

3

Fig. 8.13 Graph of P (x) (left); contours of P (x) and boundary of Uk (right). 20

10

0

-10

-20

-6

Fig. 8.14 Graphs of

-4

-2

0

2

4

6

8

P d (ς).

In two-dimensional space, if we let a11 = 3, a12 = a21 = .5, a22 = −2.0, and c11 = 1, c12 = c21 = 0, c22 = 0.5, the matrix A = {aij } is indefinite, and C = {cij } is positive definite. Setting f = {1, 1.5} and λ = 2, the graph of the canonical function P (x) = 12 xT Ax − xT f is a saddle surface (see Figure 8.13), and the boundary of the feasible set Uk = {x ∈ R2 | 12 xT Cx ≤ λ} is an ellipse (see Figure 8.13). In this case, the dual problem has four critical points (see Figure 8.14): ς¯1 = 5.22 > ς¯2 = 3.32 > ς¯3 = −2.58 > ς¯4 = −3.97. ∗ ∗ Because ς¯1 ∈ V+ and ς¯4 ∈ V− , the triality theory tells us that x1 = {−0.22, 2.81} is a global minimizer, and x4 = {−1.90, −0.85} is a local minimizer. From the graph of P d (ς) we can see that x2 = {0.59, −2.70} is a local minimizer, and x3 = {2.0, 0.15} is a local maximizer. We have

P (x1 ) = −12.44 < P (x2 ) = −4.91 < P (x3 ) = 4.03 < P (x4 ) = 9.53.


311

8.8.3 Quadratic Minimization with Box Constraints The primal problem solved in this section is finding a global minimizer of a nonconvex quadratic function over a box constraint: ½ ¾ 1 l ≤x≤ u , (8.159) (Pb ) : min P (x) = xT Ax − f T x : 2 where x ∈ Rn , and l , u are two given vectors in Rn . Problems of the form (8.159) appear frequently in partial differential equations, discretized optimal control problems, linear least squares problems, and certain successive quadratic programming methods (cf. Floudas and Visweswaran, 1995). Particularly, if l = 0 and u = 1, the problem (Pb ) is directly related to one of the fundamental problems of combinatorial optimization, namely, a continuous relaxation to the problem of minimizing a quadratic function in 0—1 variables. In order to solve this problem, we need to reformulate the constraints in canonical form. Without loss of generality, we assume that l = −1 and u = 1 (if necessary, a simple linear transformation can be used to convert the problem to this form). ½ ¾ 1 min P (x) = xT Ax − f T x : x2i ≤ 1, i = 1, . . . , n . (8.160) 2 The constraint in this problem is a vector-valued quadratic function Λ(x) = {gi (x)} = {x2i − 1} ≤ 0 ∈ Rn . Thus, the canonical dual variable ς = {ςi } should also be a vector in Rn . It has been shown recently that on the dual feasible space, Vk∗ = {ς ∈ Rn | ς ≥ 0, det (A + 2 Diag (ς)) 6= 0}, where Diag (ς) ∈ Rn×n represents a diagonal matrix with ς i , i = 1, . . . , n as its diagonal entries; the canonical dual problem is given by (see Gao, 2007a,b) ( ) n X 1 T d −1 ∗ max P (ς) = − f (A + 2 Diag (ς)) f − ςi : ς ∈ Vk . (8.161) 2 i=1 This dual problem can be solved to obtain all the critical points ¯ς . It is shown in Gao (2007a,b) that if ∗ ¯ = {ς ∈ Rn | ς ≥ 0, A + 2 Diag (ς) is positive definite}, ς ∈ V+

¯ (¯ς ) = (A + 2 Diag (¯ς ))−1 f is a global minimizer of the then the vector x primal problem.

312


8.8.4 Concave Minimization The primal problem in this case is given by (Pc ) : min{P (x) = −U (x) : Bx ≤ b, x ∈ Rn },

(8.162)

where U (x) is a convex, or even nonsmooth function, and where B ∈ Rm×n and b ∈ Rm are given. It is well known that this problem is NP-hard. Concave minimization problems constitute one of the most fundamental and intensely studied classes of problems in global minimization. A comprehensive review/survey of the mathematical properties, common applications, and solution methods is given by Benson (1995). By the use of the canonical dual transformation, a perfect dual problem has been formulated in Gao (2005a). In order to provide insights into the connection between the canonical dual transformation and the traditional Lagrange multiplier method, we demonstrate here how this perfect dual formulation can also be reproduced by the classical Lagrangian duality approach when executed in a particular fashion inspired by the canonical duality. First, let us introduce a parameter μ such that min{P (x) : Bx ≤ b} ≤ μ ≤ max{P (x) : Bx ≤ b}. Then the parameterized canonical form of this problem can be formulated as (see Gao, 2005a) (Pμ ) : min{P (x) = −U (x) : {U + μ, Bx − b} ≤ 0 ∈ R1+m , x ∈ Rn }. (8.163) In this case, the constraint g1 (x) = U (x) + μ is convex and {gi (x), i = 2, . . . , m + 1} = Bx − b are linear. By introducing Lagrange multipliers (ς, y) ∈ R1+m , and letting Va∗ = {(ς, y) ∈ R1+m | ς ≥ 0, y ≥ 0 ∈ Rm }, the Lagrangian dual to the parameterized canonical problem (8.163) is given by Ξ(x, ς, y) = (ς − 1)U (x) + μς + yT (Bx − b). Thus, by the classical Lagrangian duality, the dual problem to (Pμ ) is (LD) :

max {μς − yT b + min{(ς − 1)U (x) + yT Bx}}.

(ς,y)∈Va∗

x

(8.164)

Because U (x) is convex, the inner minimization problem in this dual form ¯ if ς > 1. has a unique solution x


313

Remark 8.1. Assume that (1) U (x) is a convex function such that x∗ = δU (x) is invertible for each x ∈ Rn , and the Legendre conjugate function U ∗ (x∗ ) = sta{xT x∗ − U (x) : δU (x) = x∗ } is uniquely defined in Rn . ¯ to the problem (Pμ ) is a KKT solution (2) An optimum solution x ¯ ≥ 0 ∈ Rm . with Lagrange multipliers ς¯ > 1, y Let ∗ V+ = {(ς, y) ∈ R1+m | ς > 1, y ≥ 0 ∈ Rm }.

Under Remark 8.1, thus, we can write (LD) in (8.164) as ½ ½ T ¾¾ y Bx T (LD) : max ∗ μς − y b + (ς − 1) min + U (x) . x ς −1 (ς,y)∈V+

(8.165)

Observe that the effect of having introduced U (x) + μ ≤ 0 is to convexity the inner minimization problem in (8.165), which, by the assumption of Remark 8.1, reduces (LD) to the following equivalent dual problem. ½ µ T ¶¾ B y (Pμd ) : max ∗ P d (ς, y) = μς − yT b + (1 − ς)U ∗ . (8.166) 1−ς (ς,y)∈V+ This is the dual problem proposed by the canonical dual transformation in Gao (2005a). By the fact that the Legendre conjugate U ∗ (x∗ ) of the convex function U (x) is also convex, this canonical dual is a concave maximization ∗ problem over the dual feasible space V+ , which can be solved uniquely for a ∗ given parameter μ ∈ R if V+ is nonempty. ¯ solves the primal problem (Pμ ) because Under Remark 8.1, note that x P (¯ x) = μ, and satisfies the KKT conditions

B¯ x ≤ b,

¯ = 0, (¯ ς − 1)δU (¯ x) + B T y

U (¯ x) + μ = 0,

T

¯ − b) = 0, ¯ (B x y

(8.167) ¯ ≥ 0, y

ς¯ > 1. (8.168)

Writing the (LD) in (8.164) as max Pθd (ς, y),

(ς,y)∈Va∗

where Pθd (ς, y) = μς − yT b + min{(ς − 1)U (x) + yT Bx}, x

we get ¯ ) = ς¯μ − bT y ¯ + (¯ ¯ T Bˆ Pθd (¯ ς, y x, ς − 1)U (ˆ x) + y T

(8.169)

ˆ satisfies δU (ˆ ¯ /(1 − ς¯). By (8.167) and the assumed invertwhere x x) = B y ˆ=x ¯ . Substituting ibility of the canonical dual relation x∗ = δU (x), we get x

314

D.Y. Gao, H.D. Sherali U∗

U x∗2 x∗1

x1

x

x∗1

x∗2

x∗

(b) Graph of the Legendre conjugate U ∗ (x∗ ).

(a) Graph of U (x).

Fig. 8.15 Nonsmooth function and its smooth Legendre conjugate.

¯ ) = P (¯ this into (8.169) and using (8.168) yields P d θ (¯ ς, y x); that is, there is zero duality gap. Furthermore, letting Uμ = {x ∈ Rn | Bx ≤ b, −U (x) = μ}, we have the following result. Theorem 8.17. (KKT Condition and Global Optimality) Under Re¯ ) ∈ Va∗ is a KKT point of (Pμd ) mark 8.1, for a given parameter μ, if (¯ ς, y such that ¯ BT y ¯∗ = x , 1 − ς¯

¯ ). ¯ = δU ∗ (¯ x∗ ) is a KKT point of (Pμ ), and P (¯ x) = P d (¯ ς, y then the vector x ∗ ¯ ¯ ) is a global maximizer of P d (ς, y) on V+ Moreover, if ς¯ > 1, then (¯ ς, y ,x is a global minimizer of P (x) on the feasible space Uμ , and min P (x) =

x∈Uμ

max P d (ς, y).

∗ (ς,y)∈V+

This example shows again that when a nonconvex constrained optimization problem can be written in a canonical form, the classical Lagrange multiplier method can be used to formulate a perfect dual problem. A detailed study on the canonical duality theory for solving general constrained nonconvex minimization problems and its connections with Lagrangian duality appears in Gao, Ruan, and Sherali (2008). One advantage of the canonical duality approach is that if the convex U (x) is nonsmooth on Ua , its Fenchel—Legendre conjugate U ∗ is a smooth function on Ua∗ (see Figure 8.15). Such an idea has also been used in the study of geometrical dual analysis for solving nonsmooth “shape-preserving” design problems (see Cheng, Fang, and Lavery, 2005, Lavery, 2004, Zhao, Fang, and Lavery, 2006).


315

8.9 Sequential Canonical Dual Transformation and Solutions to Polynomial Minimization Problems The canonical dual transformation method can be generalized in different ways to solve the global optimization problem: min{P (x) = W (x) − U (x) : x ∈ Ua }

(8.170)

with different types of nonconvex functions W (x) = V (Λ(x)) and geometrical operators Λ. If the geometrical operator Λ : U → V is a general nonlinear, nonconvex mapping, we can continue to use the canonical dual transformation such that the general nonconvex function W (x) can be written in the canonical form (see Gao, 2000a): W (x) = V (Λ(x)) = Vn (ξn (ξn−1 (. . . (ξ1 (u)) . . . ))),

(8.171)

where ξk (ξk−1 ) is either a convex or a concave function of ξk−1 , and we write Vk (ξk ) = ξk+1 (ξk ),

k = 1, . . . , n − 1.

Thus, the geometrical operator Λ : U → V in this problem is a sequential composition of nonlinear mappings Λ(k) : Vk−1 → Vk , k = 1, · · · , n, V0 = U, and Vn = V; that is, h i ξn (x) = Λ(x) = Λ(n) ◦ Λ(n−1) ◦ · · · ◦ Λ(1) (x).

Because each Vk (ξk ) is a canonical function of ξk , the canonical duality relation ςk = δVk (ξk ) : Vk → Vk∗ is one-to-one. It turns out that the Legendre conjugate Vk∗ (ςk ) = hξk ; ςk i − Vk (ξk )

can be uniquely defined. Letting ς = {ςi } ∈ Rn , the sequential canonical Lagrangian associated with the general nonconvex problem (8.170) can be written as (see Gao, 2000a) Ξ(x, ς) = hΛ(1) (x); ςn !i − Vw∗ (ς) − U (x),

(8.172)

where ςp ! := ςp ςp−1 · · · ς2 ς1 and ∗ Vw∗ (ς) = Vn∗ (ςn ) + ςn Vn−1 (ςn−1 ) + · · · +

ςn ! ∗ V (ς1 ). ς1 1

(8.173)

Thus, the canonical dual problem can be formulated as: (1)

max{P d (ς) = U Λ (ς) − Vw∗ (ς) : ς ∈ Vk∗ }.

(8.174)

316


For certain given canonical functions V , and U , and the geometrical operator Λ(1) , the Λ-conjugate transformation (1)

U Λ (ς) = sta{hΛ(1) (x); ςn !i − U (x) : δΛ(1) (x)ςn ! = δU (x)} can be well defined on certain dual feasible spaces Vk∗ , and the canonical dual variables ςk linearly depend on ς1 . This canonical dual problem can be solved very easily. Two sequential canonical dual transformation methods have been proposed in Chapter 4 of Gao (2000a). Applications to general nonconvex differential equations and chaotic dynamical systems have been given in Gao (1998a, 2000b). As an application, let us consider the following polynomial minimization problem min{P (x) = W (x) − xT f : x ∈ Rn }, (8.175)

where x = (x1 , x2 , . . . , xn )T ∈ Rn is a real vector, f ∈ Rn is a given vector, and W (x) is a so-called canonical polynomial of degree d = 2p+1 (see Gao, 2000a), defined by

W (x) =

⎛

⎛

1 ⎜1 αp ⎝ αp−1 ⎝. . . 2 2

Ã

1 α1 2

µ

1 2 |x| − λ1 2

¶2

...

!2

⎞2

⎞2

⎟ − λp−1 ⎠ − λp ⎠ ,

(8.176) where αi , λi are given parameters. It is known that the general polynomial minimization problem is NP-hard even when d = 4 (see Nesterov, 2000). Many numerical methods and algorithms have been suggested recently for finding tight lower bounds of general polynomial optimization problems (see Lasserre, 2001, Parrilo and Sturmfels, 2003). For the current canonical polynomial minimization problem, the dual problem has been formulated in Gao (2006); that is, ) ( p |f |2 X ςp ! ∗ d d (P ) : max P (ς) = − (8.177) − V (ςk ) , ς 2ςp ! ςk ! k k=1

where ς1 = ς, ςk = αk

µ

¶ 1 2 ςk−1 − λk , k = 2, . . . , p. 2αk−1

In this case, V ∗ k (ςk ) is a quadratic function of ςk defined by V ∗ k (ςk ) =

1 2 ς + λk ςk . 2αk k

(8.178)


317

The dual problem is a nonlinear program having only one variable ς ∈ R, which is much easier to solve than the primal problem. Clearly, for any ς 6= 0 and ςk2 6= 2αk λk+1 , the dual function P d is well defined and the criticality condition δP d (ς) = 0 leads to a dual algebraic equation 2 2(ςp !)2 (α−1 1 ς + λ1 ) = |f | .

(8.179)

Theorem 8.18. (Complete Solution Set to Canonical Polynomial (Gao, 2006)) For any parameters αk , and λk , k = 1, . . . , p, and input f , the dual algebraic equation (8.179) has at most s = 2p+1 − 1 real solutions: ¯ defined by ς¯(i) , i = 1, . . . , s. For each dual solution ς¯ ∈ R, the vector x ¯ (¯ x ς ) = (¯ ςp !)−1 f

(8.180)

is a critical point of the primal problem (P) and P (¯ x) = P d (¯ ς ). ¯ of the polynomial P (x) can be written in Conversely, every critical point x the form (8.180) for some dual solution ς¯ ∈ R. In the case that p = 1, the nonconvex function W (x) = 12 α1 ( 12 |x|2 − λ1 )2 is a double-well function. The global and local extrema can be identified by the triality theory given in Theorem 8.6. For the general case of p > 1, the sufficient condition for global minimizer was obtained recently in Gao (2006). Theorem 8.19. (Sufficient Condition for Global Minimizer) Suppose that for any arbitrarily given positive parameters αk , λk ≥ 0, ∀k ∈ {1, . . . , p}, ς¯ is a solution of the dual algebraic equation (8.179). If v v ⎛ ⎛ ⎞ u v u s Ã !⎞ u u u u u u 2 2 ⎜ ⎟ t 2 λp ⎠⎠, λp−1 + ς¯ > ς+ = u t2α1 ⎝λ2 + t ⎝λ3 + · · · + α2 αp−2 αp−1

then ς¯ is a global maximizer of P d on the open domain (ς+ , +∞), the vector ¯ = (¯ x ςp !)−1 f is a global minimizer of the polynomial minimization problem (8.175), and P (¯ x) = minn P (x) = max P d (ς) = P d (¯ ς ). (8.181) x∈R

ς>ς+

In the case of p = 2, the nonconvex function W (x) is a canonical polynomial of degree eight. The dual function P d (ς) has the form of µ ¶ |f |2 1 2 1 2 − ς2 + λ2 ς2 + ς2 ( ς + λ1 ς) , (8.182) Π d (ς) = − 2ςς2 α2 2α1 where ς2 = α2 ς 2 /(2α1 ) − λ2 α2 . In this case, the dual algebraic equation (8.179)

318

D.Y. Gao, H.D. Sherali 3

1.5

2

1

1 0.5 0 0 -1 -0.5

-2 -3

0

0.5

1

1.5

2

2.5

-2

3

-1

0

1

2

1

2

(a) λ1 = 0: Three solutions ς3 = 0.22 < ς2 = 1.37 < ς1 = 1.45 1.5 3 1

2 1

0.5

0 0

-1 -2

-0.5

-3 -1

0

1

2

-2

3

-1

0

(b) λ1 = 1: Five solutions {−0.96, −0.11, 0.096, 1.38, 1.45} 2 3 1.5

2 1

1

0

0.5

-1

0

-2 -0.5 -3 -2

-1

0

1

2

3

-2

-1

0

1

2

(c) λ1 = 2: Seven solutions {−2.0, −1.45, −1.35, −0.072, 0.07, 1.39, 1.44} Fig. 8.16 Graphs of the algebraic curve φ2 (ς) (left) and dual function P d (ς) (right).

2ς

2

µ

α2 2 ς − λ2 α2 2α1

¶2 µ

1 ς + λ1 α1

¶

= |f |2

(8.183)

has at most seven real roots ς¯i , i = 1, . . . , 7. Let µ ¶r α2 2 1 ς − λ2 α2 2( ς + λ1 ), φ2 (ς) = ±ς 2α1 α1 and f = {0.1, −0.1}, α1 = 1, α2 = 1, and λ2 = 1. Then, for different values of λ1 , the graphs of φ2 (ς) and P d (ς) are shown in Figure 8.16. The graphs of P (x) are shown in Figure √ 8.17 (for √λ1 = 0 and λ1 = 1) and Figure 8.18 (for λ1 = 2). Because ς+ = 2α1 λ2 = 2, √ we can see that the dual function P d (ς) is strictly concave for ς > ς+ = 2. The dual algebraic equation


319

2

2

1

1 2

0 -1

2

0 -1

0

-2

0

-2

0

0 -2

-2 2

2 (a) λ1 = 0.

(b) λ1 = 1.

Fig. 8.17 Graphs of P (x).

2 1 2 1

0 2

0

-1

-1 0

-2

-2 0 -2 2

-2

-1

0

1

2

Fig. 8.18 Graph of P (x) with λ1 = 2.

(8.183) has a total of seven real solutions when λ1 = 2, and the largest ς1 = 2.10 > ς+ = 2 gives the global minimizer x1 = f /ς1 = {2.29, −0.92}, and P (x1 ) = −1.32 = P d (ς1 ). The smallest ς7 = −4.0 gives a local maximizer x7 = {−0.04, 0.02} and P (x7 ) = 4.51 = P d (ς7 ) (see Figure 8.18). Detailed studies on solving general polynomial minimization problems are given in Gao (2000a, 2006), Lasserre (2001), and Sherali and Tuncbilek (1992, 1997).

320


8.10 Concluding Remarks We have presented a detailed review on the canonical dual transformation and its associated triality theory, with specific applications to nonconvex analysis and global optimization problems. Duality plays a key role in modern mathematics and science. The inner beauty of duality theory owes much to the fact that many different natural phenomena can be cast in the unified mathematical framework of Figure 8.1. According to the traditional philosophical principle of ying—yang duality, The Complementarity of One Ying and One Yang is the Dao (see Gao, 1996b, Lao Zhi, 400 BC); that is, the constitutive relations in any physical system should be one-to-one. Niels Bohr realized its value in quantum mechanics. His complementarity theory and philosophy laid a foundation on which the field of modern physics was developed (Pais, 1991). In nonconvex analysis and optimization, this one-to-one canonical duality relation serves as the foundation for the canonical dual transformation method. For any given nonconvex problem, as long as the geometrical operator Λ is chosen properly and the tricanonical forms can be characterized correctly, the canonical dual transformation can be used to establish elegant theoretical results and to develop efficient algorithms for robust computations. The extended Lagrangian duality and triality theories show promise of having significance in many diverse fields. As indicated in Gao (2000a), duality in natural systems is a very broad and rich field. To theoretical scientists and philosophical thinkers as well as great artists, duality has always played a central role in their respective fields. It is really “a splendid feeling to realize the unity of a complex of phenomena that by physical perception appear to be completely separated” (Albert Einstein). It is pleasing to see that more and more knowledgeable researchers and scientists are working in this wonderland and exploring the intrinsic beauty of nature, often revealed via duality theory. Acknowledgments This work is supported by the National Science Foundation by Grant Numbers DMII-0455807, CCF-0514768, and DMII-0552676.

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Chapter 9

Quantum Computation and Quantum Operations Stan Gudder

Summary. Quantum operations play an important role in quantum measurement, quantum computation, and quantum information theories. We classify quantum operations according to certain special properties such as unital, tracial, subtracial, self-adjoint, and idempotent. We also consider a type of quantum operation called a L¨ uders map. Examples of quantum operations that describe noisy quantum channels are discussed. Results concerning iterations and fixed points of quantum operations are presented. The relationship between quantum operations and completely positive maps is discussed and the sequential product of quantum effects is considered. Key words: Quantum computation, quantum operation, quantum channel, quantum information theory

9.1 Introduction and Basic Definitions The main arena for studies in quantum computation and quantum information is a finite-dimensional complex Hilbert space which we denote by H. We denote the set of bounded linear operators on H by B(H) and we use the notation B(H)+ = {A ∈ B(H) : A ≥ 0} E(H) = {A ∈ B(H) : 0 ≤ A ≤ I} © ª D(H) = ρ ∈ B(H)+ : tr(ρ) = 1 . Stan Gudder Department of Mathematics, University of Denver, Denver, Colorado 80208 e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_9, © Springer Science+Business Media, LLC 2009

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Stan Gudder

The elements of E(H) are called effects and the elements of D(H) are called states (or density operators). It is clear that D(H) ⊆ E(H) ⊆ B(H)+ . Effects correspond to quantum yes—no measurements that may be unsharp. If a quantum system is in the state ρ, then the probability that the effect A occurs (has answer yes) is given by Pρ (A) = tr(ρA). As we show, quantum measurements with more than two possible values (not just yes—no) can be described by quantum operations. It is easy to check that D(H) forms a convex subset of B(H) and the extreme points of D(H) are called pure states. The pure states have the form Pψ where Pψ denotes a one-dimensional projection onto a unit vector ψ ∈ H. If ρ = Pψ is a pure state, then Pρ (A) = tr(Pψ A) = hAψ, ψi. ∗ Let Ai ∈ B(H), i = 1, . . . , n, and let A = {Ai , A Pi : i = 1,∗ . . . , n}. We call the map φA : B(H) → B(H) given by φA (B) = Ai BAi a quantum operation and we call the operators Ai , i = 1, . . . , n, the operation elements of φA . Notice that φA : B(H)+ → B(H)+ ; that is, φA preserves positivity. Also, φA is linear and A ≤ B implies that φA (A) ≤ φA (B).PWe say that φ Ai A∗i = I, PA is∗ unital, tracial, P ∗or subtracial, respectively, in the case Ai Ai = I, or Ai Ai ≤ I, respectively. Notice that φA is a unital if and only if φA (I) = I, φA is tracial if and only if tr (φA (B)) = tr(B) for every B ∈ B(H), and φA is subtracial if and only if tr (φA (B)) ≤ tr(B) for every B ∈ B(H)+ . We say that φA is self-adjoint if Ai = A∗i , i = 1, . . . , n. An important type of self-adjoint quantum operation in quantum measurement P 1/2 1/2 theory [4, 7,P 9] is a L¨ uders map of the form L(B) = Ai BAi where Ai ∈ E(H) with Ai = I, i = 1, . . . , n. In this case, L is unital and tracial and {Ai : i = 1, . . . , n} is called a finite POV (positive operator-valued ) measure. We interpret the POV measure {Ai : i = 1, . . . , n} as a quantum measurement with n possible values (which can be taken to be 1, . . . , n). Restricting L to E(H) we have L : E(H) → E(H) and L(B) is interpreted as the effect resulting from first making the measurement described by {Ai : i = 1, . . . , n} and then measuring B. If we restrict L to D(H) then L : D(H) → D(H) is called the square root dynamics [2]. Quantum operations have various interpretations in quantum measurement, computation, and information theories [1, 4, 7, 8, 9, 10]. If φA is tracial, then φA : D(H) → D(H) can be thought of as a quantum measurement with possible outcomes 1, 2, . . . , n. If the measurement is performed on a quantum system in the state ρ ∈ D(H), then the probability of obtaining outcome i is tr(Ai ρA∗i ) and the postmeasurement state given that i occurs is Ai ρA∗i /tr(Ai ρA∗i ). Moreover, the resulting state after the measurement is executed but no observation is made is given by φA (ρ). Quantum operations can also be interpreted as an interaction of a quantum system with an environment followed by a unitary evolution, a noisy quantum channel, or a quantum error correction map [10]. Depending on the application, at least

9 Quantum Computation and Quantum Operations

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one of our previous properties is assumed to hold. For illustrative purposes, we mainly consider the noisy quantum channel interpretation. Notice that if φA and φB© are quantum operations on B(H) with A = ª {Ai , A∗i : i = 1, . . . , n}, B = Bj , Bj∗ : j = 1, . . . , m , then their composition φA ◦ φB is a quantum operation on B(H) with operation elements Ai Bj , i = 1, . . . , n, j = 1, . . . , m. If A = B we write φ2A = φA ◦ φA , . . . , φnA = φA ◦ · · · φA

(n factors).

A quantum operation φA is idempotent if φ2A = φA . We now give some simple basic results. Lemma 9.1.1. If φA and φB are both unital, tracial, or subtracial, respectively, then φA ◦ φB is unital, tracial, or subtracial, respectively. P P Proof. If φA and φB are both unital, then Ai A∗i = Bj Bj∗ = I. Hence, X

Ai Bj (Ai Bj )∗ =

i,j

X

Ai Bj Bj∗ A∗i =

i,j

=

X

X

Ai

i

Ai A∗i

X

Bj Bj∗ A∗i

j

= I.

i

Therefore, φA ◦ φB is unital. In a similar way, if φA and φB are both tracial then φA ◦ φB is tracial. Now suppose that P φ∗A and φB are both subtracial. Then there exists a C ∈ E(H) such that Ai Ai + C = I. Hence, X X X X (Ai Bj )∗ Ai Bj = Bj∗ A∗i Ai Bj = Bj∗ A∗i Ai Bj i,j

i,j

=

X

Bj∗ Bj

Therefore, φA ◦ φB is subtracial.

−

X

i

Bj∗ CBj

j

≤

X

Bj∗ Bj ≤ I.

Lemma 9.1.2. If φA is subtracial and its operation elements are self-adjoint projection operators, then φA is idempotent. P P Proof. We have that φA (B) = Ai BAi where Ai = A∗i = A2i and Ai ≤ I, i = 1, . . . , n. For i, j ∈ {1, . . . , n}, i 6= j, we have X Ai + Aj ≤ Ak ≤ I. It follows that Ai Aj = Aj Ai = 0 for i 6= j. Hence, X X φA ◦ φA (B) = Aj Ai BAi Aj = Ai BAi = φA (B) i,j

so that φA is idempotent.

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9.2 Completely Positive Maps In Section 9.1 we defined a quantum operation as a map φ : B(H) → B(H) of the form X φ(B) = Ai BA∗i (9.1)

and in Section 9.3 we give some simple practical examples of quantum operations. But why do quantum operations have the operator-sum form (9.1)? The present section tries to answer this question in terms of completely positive maps. We can consider Mk = B(Ck ) as the set of all k × k complex matrices, k = 1, 2, . . . . The set of operators in the tensor product B(H) ⊗ Mk = B(H ⊗ Ck ) can be considered to be the set of k × k matrices with entries in B(H). For example if A, B, C, D ∈ B(H), then the matrix ∙ ¸ AB M= CD is an element of B(H) ⊗ M2 . Of course, M ∈ B(H ⊗ C2 ) in the sense that ∙ ¸ ∙ ¸ x Ax + By M = y Cx + Dy for all x, y ∈ H. For a linear map φ : B(H) → B(H) we define the linear maps φk : B(H) ⊗ Mk → B(H) ⊗ Mk given by φk (M ) = [φ(Mij )] , where M = [Mij ] ∈ B(H)⊗Mk , i, j = 1, . . . , k. If φk sends positive operators into positive operators for k = 1, 2, . . . , then φ is called completely positive. It is easy to check that φ : B(H) → B(H) is completely positive if and only if φ ⊗ Ik : B(H) ⊗ Mk → B(H) ⊗ Mk preserves positivity for k = 1, 2, . . . , where Ik is the identity map on Mk . We have seen that a quantum operation φ : B(H) → B(H) describes various ways that states are transformed into other states for a quantum system. Because states are positive operators, φ must preserve positivity. Now suppose our quantum system interacts (or couples) with an environment such as a noisy quantum channel. If this environment is described by the Hilbert space Ck , then the combined system is described by the tensor product H ⊗ Ck . The natural extension of φ to the combined system is given by φ ⊗ Ik : B(H ⊗ Ck ) → B(H ⊗ Ck ). The map φ ⊗ Ik just acts on B(H) like φ and leaves the environment unaltered. We would expect φ ⊗ Ik to map states into states so φ⊗Ik should also preserve positivity, k = 1, 2, . . . . We conclude that quantum operations should be completely positive maps. If x, y ∈ H we define the linear operator |xihy| ∈ B(H) by |xihy|v = hy, vix for every v ∈ H. If {x1 , . . . , xn } is an orthonormal basis for H, then any


A ∈ B(H) has the form

A=

X

331

aij |xi ihxj |,

(9.2)

where aij ∈ C, i, j = 1, . . . , n. Now let {y1 , . . . , yk } be an orthonormal basis for Ck . Then an orthonormal basis for H ⊗ Ck is given by {xi ⊗ yj : i = 1, . . . , n; j = 1, . . . , k} . For an operator M ∈ B(H ⊗ Ck ) as in (9.2) we have X M = ar,s,i,j |xr ⊗ yi ihxs ⊗ yj | r,s,i,j

=

X

r,s,i,j

=

Ã X X r,s

i,j

=

ar,s,i,j |xr ihxs | ⊗ |yi ihyj |

X i,j

!

ar,s,i,j |xr ihxs |

⊗ |yi ihyj |

Ai,j ⊗ |yi ihyj |,

where Aij =

X r,s

(9.3)

arsij |xr ihxs | ∈ B(H).

If φ : B(H) → B(H) is a linear map and M ∈ B(H⊗Ck ) has the representation (9.3), then φ ⊗ Ik : B(H ⊗ Ck ) → B(H ⊗ Ck ) satisfies X (φ ⊗ Ik )(M ) = φ(Aij ) ⊗ |yi ihyj |. (9.4) i,j

The following structure theorem is due to Choi [6]. Theorem 9.2.1. A linear map φ : B(H) → B(H) is completely positive if and only if there exist a finite number of operators Ai ∈ B(H) such that (9.1) holds for every B ∈ B(H). Proof. Suppose φ has the representation (9.1). Applying (9.4) we have X (φ ⊗ Ik )(M ) = φ(Aij ) ⊗ |yi ihyj | i,j

=

XX i,j

r

Ar Aij A∗r ⊗ |yi ihyj |.

Now any z ∈ H ⊗ Ck can be represented in the form X z= us ⊗ vs ,

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where us ∈ H, vs ∈ Ck . Writing zr =

P

h(φ ⊗ Ik )(M )z, zi =

∗ s Ar us

X r

⊗ vs it is easy to check that

hM zr , zr i ≥ 0

because M is positive. Conversely, let φ : B(H) → B(H) be a completely positive map. Let {x1 , . . . , xn } and {y1 , . . . , yn } be two orthonormal bases for H. Now φ ⊗ In is positivity preserving. The operator M ∈ B(H ⊗ H) defined by X M = |xr ihxs | ⊗ |yr ihys | r,s

=

X r,s

|xr ⊗ yr ihxs ⊗ ys |

¯ ¯ +* ¯ ¯X X ¯ ¯ xr ⊗ yr xs ⊗ ys ¯ =¯ ¯ ¯ r s

is positive because M is a multiple of a one-dimensional projection. Hence, X (φ ⊗ In )(M ) = φ (|xr ihxs |) ⊗ |yr ihys | (9.5) r,s

is a positive operator. By the spectral theorem there exists an orthonormal basis {v1 , . . . , vm } of H ⊗ H where m = n2 and positive numbers λ1 , . . . , λm such that ¯ EDp X X ¯¯p ¯ (φ ⊗ In )(M ) = (9.6) λi |vi ihvi | = λi vi ¯. ¯ λi vi r,s

P If v = vij xi ⊗ yj is a vector in H ⊗ H we associate with v an operator Av ∈ B(H) by X Av = |xi ihxj |. (9.7) i,j

Then a straightforward computation gives X |vihv| = Av |xr ihxs |A∗v ⊗ |yr ihys |.

(9.8)

r,s

√ Associating with each λi vi in (9.6) the operator Ai in (9.7) and using (9.8) we have X (φ ⊗ In )(M ) = Ai |xr ihxs |A∗i ⊗ |yr ihys |. (9.9) i,r,s

Applying (9.5) and (9.9) gives


φ (|xr ihxs |) =

X i

333

Ai |xr ihxs |A∗i .

Because the operators |xr ihxs | span the whole space B(H), we conclude that (9.1) holds for every B ∈ B(H). We now show that the operator-sum representation (9.1) is not unique. In other words, the operation elements for a quantum operation are not unique. Let φ and ψ be quantum operations acting on B(C2 ) with operation-sum representations φ(B) = E1 BE1∗ + E2 BE2∗ ψ(B) = F1 BF1∗ + F2 BF2∗ , where ∙ ¸ 1 10 E1 = √ 2 01 ∙ ¸ 10 F1 = 00

∙ ¸ 1 1 0 E2 = √ 2 0 −1 ∙ ¸ 00 F1 = . 01

Although φ and ψ appear to be quite different, they are actually the same quantum operation. To see this, note that F1 = √12 (E1 + E2 ) and F2 = √1 (E1 − E2 ). Thus, 2 ψ(B) =

(E1 + E2 )B(E1 + E2 ) + (E1 − E2 )B(E1 − E2 ) 2

= E1 BE1 + E2 BE2 = φ(B). Notice that in the previous example we could write Fi = [uij ] is the unitary matrix ∙ ¸ 1 1 1 √ . 2 1 −1

P

uij Ej where

In this sense, the operation elements of ψ are related to the operation elements of φ by a unitary matrix. The next theorem, whose proof may be found in [10], shows that this holds in general. Theorem 9.2.2. Suppose {E1 , . . . , En } and {F1 , . . . , Fm } are operation elements giving rise to subtracial quantum operations φ and ψ, respectively. By appending zero operators to the shorter list of operation elements we may assume that m = n. PThen φ = ψ if and only if there exist complex numbers uij such that Fi = j uij Ej where [uij ] is an m × m unitary matrix. This theorem is important in the development of quantum error-correcting codes [10]. Suppose we have two representations

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φ(B) = for the quantum operation φ.

X

Ei BEi∗ =

X

Fj BFj∗

Lemma 9.2.3. The quantum operation φ is unital, tracial, or subtracial, respectively, with respect to the operation elements {E1 , . . . , En } if and only if φ is unital, tracial, or subtracial, respectively, with respect to the operation elements {F1 , . . . , Fm }. Proof. If φ is unital with respect to {E1 , . . . , En }, then X X Fj Fj∗ = φ(I) = Ei Ei∗ = I

so φ is unital with respect to {F1 , . . . , Fm }. If φ is tracial with respect to {F1 , . . . , Fm }, then for any B ∈ B(H) we have ³X ´ ´ ³X tr(B) = tr (φ(B)) = tr Fj BFj∗ = tr Fj∗ Fj B . P ∗ It follows that Fj Fj = I so φ is tracial with respect to {F1 , . . . , Fm }. The subtracial proof is similar.

This last lemma P does not apply to self-adjoint quantum operations. For example,Pif φ(B) = Aj BA∗j where the Aj are self-adjoint we can also write φ(B) = (iAj )B(iAj )∗ where iAj are not self-adjoint. We now give an example which shows that a positivity preserving map need not be completely positive. Define φ : B(C2 ) → B(C2 ) by φ(A) = AT where AT is the transpose of A. Now a matrix ∙ ¸ ab A= ∈ B(C2 ) cd is positive if and only if a ≥ 0, d ≥ 0, and ad − bc ≥ 0. Hence, if A ≥ 0 then AT ≥ 0 so φ is positivity preserving. To show that φ is not completely positive consider φ ⊗ I2 on B(C2 ⊗ C2 ). Let ei = (1, 0), e2 = (0, 1) be the standard basis for C2 and define the positive operator A ∈ B(C2 ⊗ C2 ) by A = |e1 ⊗ e1 + e2 ⊗ e2 ihe1 ⊗ e2 + e2 ⊗ e2 | = |e1 ⊗ e1 ihe1 ⊗ e1 | + |e1 ⊗ e1 ihe2 ⊗ e2 | + |e2 ⊗ e2 ihe1 ⊗ e1 | + |e2 ⊗ e2 ihe2 ⊗ e2 | = |e1 ihe1 | ⊗ |e1 ihe1 | + |e1 ihe2 | ⊗ |e1 ihe2 | + |e2 ihe1 | ⊗ |e2 ihe1 | + |e2 ihe2 | ⊗ |e2 ihe2 | . We then have


335

(φ ⊗ I2 )(A) = |e1 ihe1 | ⊗ |e1 ihe1 | + |e2 ihe1 | ⊗ |e1 ihe2 | + |e1 ihe2 | ⊗ |e2 ihe1 | + |e2 ihe2 | ⊗ |e2 ihe2 | = |e1 ⊗ e1 ihe1 ⊗ e1 | + |e1 ⊗ e1 ihe1 ⊗ e2 | + |e1 ⊗ e2 ihe2 ⊗ e1 | + |e2 ⊗ e2 ihe2 ⊗ e2 | ⎤ 1000 ⎢0 0 1 0⎥ ⎥ =⎢ ⎣0 1 0 0⎦. 0001 ⎡

But letting v = (0, 1, −1, 0) ∈ C2 ⊗ C2 we have

h(φ ⊗ Ir )(v), vi = h(0, −1, 1, 0), (0, 1, −1, 0)i = −2. Hence, φ ⊗ I2 is not positivity preserving so φ is not completely positive.

9.3 Noisy Quantum Channels This section discusses the quantum operation descriptions of some simple noisy quantum channels [10]. A two-dimensional quantum system is called a qubit. This is the most basic quantum system studied in quantum computation and quantum information theory. A qubit has a two-dimensional state space C2 with (computational) basis elements |0i = (1, 0) and |1i = (0, 1). The bit flip channel flips the state of a qubit from |0i to |1i (and vice versa) with probability 1 − p, 0 < p < 1. Letting X be the Pauli matrix ∙ ¸ 01 X= , 10 we can represent the bit flip channel by the quantum operation φbf (ρ) = pρ + (1 − p)XρX. © ª Notice that φbf has operation elements p1/2 I, (1 − p)1/2 X and that φbf is self-adjoint and tracial. It is also unital because for any self-adjoint quantum operation tracial and unital are equivalent. Of course, φbf gives a bit flip because X|0i = |1i and X|1i = |0i. Hence, φbf (|0ih0|) = p|0ih0| + (1 − p)|1ih1| so the pure state |0ih0| is left undisturbed with probability p and is flipped with probability 1 − p. Similarly,

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φbj (|1ih1|) = p|1ih1| + (1 − p)|0ih0| . The phase flip channel is represented by the quantum operation φpf (ρ) = pρ + (1 − p)ZρZ, where 0 < p < 1 and Z is the Pauli matrix ∙ ¸ 1 0 Z= . 0 −1

© ª The operation elements for φpf are p1/2 I, (1 − p)1/2 Z so again φpf is selfadjoint and tracial. Because Z|0i = |0i and Z|1i = −|1i we see that φpf changes the relative phase of the qubit states with probability 1 − p. The bit-phase flip channel is represented by the quantum operation φbpf (ρ) = pρ + (1 − p)Y ρY, where 0 < p < 1 and Y is the Pauli matrix ∙ ¸ 0 −i Y = . i 0 This gives a combination of a bit © flip and a phase flipªbecause Y = iXZ. The operation elements for φbpf are p1/2 I, (1 − p)1/2 Y so φbpf is self-adjoint and tracial. We obtain an interesting quantum operation by forming the composition φbf ◦ φpf . Because XZ = −iY we have φbf ◦ φpf (ρ) = p2 ρ + p(1 − p)ZρZ + p(1 − p)XρX + (1 − p)2 Y ρY. The operation elements become n p o p pI, p(1 − p) Z, p(1 − p) X, (1 − p)Y

so again, φbf ◦ φpf is self-adjoint and tracial. It is also easy to check that φpf ◦ φbf = φbf ◦ φpf . Another important type of quantum noise is the depolarizing channel given by the quantum operation φdp (ρ) =

pI + (1 − p)ρ, 2

where 0 < p < 1. This channel depolarizes a qubit state with probability p. That is, the state ρ is replaced by the completely mixed state I/2 with probability p. By applying the identity ρ + XρX + Y ρY + ZρZ I = 2 4


337

that holds for every ρ ∈ D(C2 ) we can write µ ¶ 3 p φdp (ρ) = 1 − p ρ + (XρX + Y ρY + ZρZ) . 4 4 Thus, the operation elements for φdp become o np √ √ √ 1 − 3p/4 I, p X/2, p Y /2, p Z/2 .

As before, φdp is self-adjoint and tracial. There are practical quantum operations that are not self-adjoint or unital. For example, consider the amplitude damping channel given by the quantum operation φad (ρ) = A1 ρA∗1 + A2 ρA∗2 , where

∙

1√ 0 A1 = 0 1−γ

¸

,

∙ √ ¸ 0 γ , A2 = 0 0

and 0 < γ < 1. It is easy to check that φad is tracial but not self-adjoint nor unital. Although the quantum channels (quantum operations) that we have considered appear to be quite specialized, general quantum channels and quantum operations can be constructed in terms of these simple ones and this is important for the theory of quantum error correction.

9.4 Iterations It is sometimes important to consider iterations of quantum operations. For example, a measurement may be repeated many times for greater accuracy or quantum data may enter a noisy channel several times. For a quantum operation φA , does the sequence of iterations φnA (ρ), n = 1, 2, . . . , converge for every state ρ ∈ D(H)? (Because H is finite-dimensional, all the usual forms of convergence such as norm convergence or matrix entry convergence coincide so we do not need to specify a particular type of convergence.) In general, the answer is no. For example, φ(ρ) = XρX is a self-adjoint, tracial, and unital quantum operation. Because X 2 = I we have φ2n (ρ) = ρ, n = 1, 2, . . . , but φ2n+1 (ρ) = XρX, n = 1, 2, . . . . Unless ρX = Xρ, the sequence of iterates does not converge. A state ρ0 is a fixed point of a quantum operation φA if φA (ρ0 ) = ρ0 . It is frequently useful to know the fixed points of a quantum operation because these are the states that are not disturbed by a quantum measurement or a noisy quantum channel. Lemma 9.4.1. A state ρ0 is a fixed point of φA if and only if there exists a state ρ such that lim φnA (ρ) = ρ0 .

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Proof. If lim φnA (ρ) = ρ0 , by the continuity of φA we have that n n ρ0 = lim φn+1 A (ρ) = lim φA ◦ φA (ρ) = φA (lim φA (ρ)) = φA (ρ0 ).

Hence, ρ0 is a fixed point of φA . Conversely, if ρ0 is a fixed point of φA we have that φnA (ρ0 ) = φn−1 (φA (ρ0 )) = φn−1 A A (ρ0 ) = · · · = φA (ρ0 ) = ρ0 . Hence, lim φnA (ρ0 ) = ρ0 .

The next result shows that the iterates of some of the quantum operations considered in Section 9.3 always converge. Theorem 9.4.2. For any ρ ∈ D(C2 ) we have that (a) lim φnbf (ρ) = 12 ρ + 12 XρX (b) lim φnpf (ρ) = 12 ρ + 12 ZρZ (c) lim φnbpf (ρ) = 12 ρ + 12 Y ρY (d) lim φndp (ρ) = I2 .

Proof. (a) Any ρ ∈ D(C2 ) has the Bloch form ¸ ∙ 1 1 + r3 r1 − ir2 ρ= , 2 r1 + ir2 1 − r3 where ri ≥ 0, i = 1, 2, 3, and r12 + r22 + r32 ≤ 1. Because ¸ ∙ 1 1 − r3 r1 + ir2 XρX = 2 r1 − ir2 1 + r3 we have that ¸ ∙ 1 1 + (2p − 1)r3 r1 − i(2p − 1)r2 φbf (ρ) = . 2 r1 + i(2p − 1)r2 1 − (2p − 1)r3 We can now prove by induction that ¸ ∙ 1 1 + (2p − 1)n r3 r1 − i(2p − 1)n r2 φnbf (ρ) = . 2 r1 + i(2p − 1)n r2 1 − (2p − 1)n r3 Because 0 < p < 1, we have −1 < 2p − 1 < 1 so that lim(2p − 1)n = 0. Hence, ¸ ∙ 1 1 r1 1 1 lim φnbf (ρ) = = ρ + XρX. 2 r1 1 2 2 The proofs of (b) and (c) are similar. To prove (d), a simple induction argument shows that for every ρ ∈ D(C2 ) φndp (ρ) =

1 − qn I + (1 − p)n ρ, 2


339

where q = 1 − p. Because 0 < q, p < 1, we have that lim φnbf (ρ) =

I . 2 1/2

We see from Theorem 9.4.2(a) that lim φnbf = φbf where 1/2

φbf (ρ) =

1 1 ρ + XρX 2 2 1/2

and similar results hold for φpf and φbpf . Notice that φbf is an idempotent quantum operation. Indeed, 1/2

1/2

1 ρ+ 4 1 = ρ+ 2

φbf ◦ φbf (ρ) =

1 1 1 XρX + XρX + X 2 ρX 2 4 4 4 1 1/2 XρX = φbf (ρ) . 2

The next result shows that this always happens. Theorem 9.4.3. If there exists a quantum operation φ such that lim φnA (ρ) = φ(ρ) for every ρ ∈ D(H), then φ is idempotent. Moreover, the set of fixed points of φA coincides with the range ran(φ). Proof. By the continuity of φnA we have ´ ³ m+n (ρ) = φ(ρ) . φnA (φ(ρ)) = φnA lim φm A (ρ) = lim φA m→∞

m→∞

Hence,

φ ◦ φ(ρ) = lim φnA (φ(ρ)) = φ(ρ) and we conclude that φ is idempotent. The last statement follows from Lemma 9.4.1.

9.5 Fixed Points Let φA be a quantum operation with A = {Ai , A∗i : i = 1, . . . , n}. The commutant A0 of A is the set A0 = {B ∈ B(H) : BAi = Ai B, BA∗i = A∗i B, i = 1, . . . , n} . We denote the set of fixed states of φA by I(φA ). That is, I(φA ) = {ρ ∈ D(H) : φA (ρ) = ρ} . As an example, it is easy to find I(φpf ). In this case ρ ∈ I(φpf ) if and only if ρ = pρ + (1 − p)ZρZ. This is equivalent to ρ = ZρZ. Because Z 2 = I we

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have that Zρ = ρZ. We conclude that ρ ∈ I(φpf ) if and only if ρ ∈ A0 where A = {I, Z}. A similar result holds for φbf and φbpf . In general we have the following result which is a special case of a theorem in [1, 5]. Theorem 9.5.1. If φA is a self-adjoint, subtracial quantum operation, then I(φA ) ⊆ A0 ∩ D(H). Proof. Let ρ ∈ I(φA ) and let h be a unit eigenvector of ρ corresponding to the largest eigenvalue λ1 = kρk. Then φA (ρ) = ρ implies that X X X ® λ1 = A2i h, h ≤ λ1 . hρAi h, Ai hi ≤ kρk kAi hk2 = λ1 ® Because hρAi h, Ai hi ≤ λ1 A2i h, h , it follows that

h(λ1 I − ρ)Ai h, Ai hi = 0.

Hence, (λ1 I −ρ)Ai h = 0 for every eigenvector h corresponding to λ1 . Thus, Ai leaves the λ1 -eigenspace invariant. Letting P1 be the corresponding spectral projection of ρ we have P1 Ai P1 = Ai P1 which implies that Ai P1 = P1 Ai , i = 1, . . . , n. Now ρ = λ1 P1 + ρ1 where ρ1 is a positive operator with largest eigenvalue. Because λ1 P1 + ρ1 = ρ = φA (ρ) = λ1 φA (P1 ) + φA (ρ1 ) = λ1 P1 + φA (ρ1 ) we have φA (ρ1 ) = ρ1 . Proceeding by induction, ρ ∈ A0 . Corollary 9.5.2. If φA is a self-adjoint, tracial quantum operation, then I(φA ) = A0 ∩ D(H) . As an application of Corollary 9.5.2 we see that I(φdp ) = {I/2}. Indeed, if ρ = I(φA ) then ρ must commute with X,Y , and Z. But any 2 × 2 matrix is a linear combination of I, X, Y , and Z. It follows that ρ = I/2. The next example which is a special case of an example in [3] shows that self-adjointness cannot be deleted Pfrom Theorem 9.5.1 or Corollary 9.5.2. Let φA (B) = 4i=1 Ai BA∗i be the quantum operation with ⎡ ⎤ ⎡ ⎤ 100 000 A1 = ⎣ 0 0 0 ⎦ , A2 = ⎣ 0 1 0 ⎦ , 000 000 ⎡ ⎤ ⎡ ⎤ 000 000 1 ⎣ 1 0 0 0⎦, A3 = √ A4 = √ ⎣ 0 0 0 ⎦ . 2 100 2 010

It is easy to check that φA is unital. However, φA is not self-adjoint and because


X

341

⎡

⎤

100 3⎣ 0 1 0⎦ 2 000

A∗i Ai =

we see that φA is not subtracial. Let ρ ∈ D(C3 ) be the state ⎡ ⎤ 200 1 ρ = ⎣0 0 0⎦. 3 001

/ A0 . If we It is easy to check that ρ ∈ I(φA )pbut ρA3 6= A3 ρ so that ρ ∈ multiply the Ai , i = 1, 2, 3, 4, by 2/3 then φA would be subtracial but again I(φA ) 6⊆ A0 ∩ D(H).

9.6 Idempotents We showed in Lemma 9.1.2 that if φA is subtracial and its operation elements are self-adjoint projection operators, then φA is idempotent. We conjecture that a weak converse of this result holds. If L is a L¨ uders map that is idempotent, we conjecture that L can be written in a form so that its operation elements are self-adjoint projections. As a start, our next result shows that this conjecture holds in C2 if L has two operation elements. 1/2

1/2

1/2

1/2

Theorem 9.6.1. Suppose L(B) = A1 BA1 + A2 BA2 , A1 , A2 ≥ 0, A1 + A2 = I, is a L¨ uders map on C2 and L2 = L. Then A1 and A2 are self-adjoint projection operators or L is the identity map. Proof. Because A1 + A2 = I, A1 and A2 commute and because L2 = L we have 1/2

1/2

1/2

1/2

A1 BA1 + A2 BA2

1/2

1/2

1/2

1/2

= A1 BA1 + A2 BA2 + 2A1 A2 BA2 A1

(9.10)

for every B ∈ B(C2 ). Without loss of generality, we can assume that A1 is diagonal so that ∙ ¸ a0 A1 = , 0 ≤ a, b ≤ 1. 0b Letting

∙

11 B= 11

¸

in equation (9.10) and equating entries we obtain p p √ ab + (1 − a)(1 − b) = (1 − a)(1 − b) + ab + 2 ab(1 − a)(1 − b) . (9.11) Equation (9.11) can be written as

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´ ³√ ´ ³ p p √ ab + (1 − a)(1 − b) = 0. 1 − ab − (1 − a)(1 − b)

p √ We conclude that ab + (1 − a)(1 − b) = 0 or 1. In the first case a = 0, b = 1 or a = 1, b = 0 and we are finished. In the second case, we can square the expression to obtain p 2 ab(1 − a)(1 − b) = a + b − 2ab. (9.12)

Squaring (9.12) gives

(a − b)2 = a2 + b2 − 2ab = 0 so that a = b. Hence, A1 = aI, A2 = (1 − a)I, and L(B) = B for all B ∈ B(C2 ).

9.7 Sequential Measurements This section discusses a topic that is important in quantum measurement theory, namely sequential products of effects. In this section we allow H to be infinite-dimensional and again denote the set of effects on H by E(H). Recall that effects represent yes—no quantum measurements that may be unsharp (imprecise). We may think of effects as fuzzy quantum events. Sharp quantum events are represented by self-adjoint projection operators. Denoting this set by P(H) we have that P(H) ⊆ E(H). We mentioned in Section 9.1 that for a quantum system initially in the state ρ ∈ D(H), the postmeasurement state given that A ∈ E(H) occurs is A1/2 ρA1/2 /tr(ρA). Assuming that tr(ρA) 6= 0, it is reasonable to define the conditional probability of B ∈ E(H) given A ∈ E(H) to be Pρ (B | A) =

tr(A1/2 ρA1/2 B) tr(ρA1/2 BA1/2 ) = . tr(ρA) tr(ρA)

(9.13)

Now two measurements A, B ∈ E(H) cannot be performed simultaneously in general, so they are frequently executed sequentially. We denote by A ◦ B a sequential measurement in which A is performed first and B second. It is natural to assume the probabilistic equation Pρ (A ◦ B) = Pρ (A)Pρ (B | A) .

(9.14)

Combining (9.13) and (9.14) gives tr(ρA ◦ B) = tr(ρA1/2 BA1/2 ) .

(9.15)


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Equation (9.15) motivates our definition A ◦ B = A1/2 BA1/2 and we call A ◦ B the sequential product of A and B. If {A1 , . . . , An } is a finite POV measure, then uders map with operation elements Ai can now be written P the L¨ as L(B) = Ai ◦ B. Notice that A ◦ B ∈ E(H) so ◦ gives a binary operation on E(H). Indeed, D E D E D E 0 ≤ A1/2 BA1/2 x, x = BA1/2 x, A1/2 x ≤ A1/2 x, A1/2 , x (9.16) = hAx, xi ≤ hx, xi so that 0 ≤ A1/2 BA1/2 ≤ I. It also follows from (9.16) that A ◦ B ≤ A. We say that A, B ∈ E(H) are compatible if AB = BA. It is clear that the sequential product satisfies 0◦A=A◦0=0 I ◦A=A◦I =A A ◦ (B + C) = A ◦ B + A ◦ C whenever B + C ≤ I (λA) ◦ B = A ◦ (λB) = λ(A ◦ B) for 0 ≤ λ ≤ 1. However, A ◦ B has practically no other algebraic properties unless compatibility conditions are imposed. To illustrate the fact that A ◦ B does not have properties that one might expect, we now show that A ◦ B = A ◦ C does not imply that B ◦ A = C ◦ A even for A, B, C ∈ P(H). In H = C2 consider A, B, C ∈ P(H) given by the following matrices, ∙ ¸ ∙ ¸ ∙ ¸ 1 11 10 00 A= , B= , C= . 00 01 2 11 We then have A ◦ B = ABA =

1 A = ACA = A ◦ C. 2

However, 1 1 B 6= C = CAC = C ◦ A. 2 2 This example also shows that A ◦ B 6≤ B in general, even though we always have A ◦ B ≤ A. We say that A, B are sequentially independent if A ◦ B = B ◦ A. It is clear that if A and B are compatible, then they are sequentially independent. To prove the converse, we need the following result due to Fuglede—Putnam— Rosenblum [11]. B ◦ A = BAB =

Theorem 9.7.1. If M, N, T ∈ B(H) with M and N normal, then M T = T N implies that M ∗ T = T N ∗ . Corollary 9.7.2. [8] For A, B ∈ E(H), A ◦ B = B ◦ A implies AB = BA. Proof. Because A ◦ B = B ◦ A we have

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A1/2 B 1/2 B 1/2 A1/2 = B 1/2 A1/2 A1/2 B 1/2 . Hence, M = A1/2 B 1/2 and N = B 1/2 A1/2 are normal. Letting T = A1/2 we have M T = T N . Applying Theorem 9.7.1, we conclude that B 1/2 A = AB 1/2 . Hence, BA = B 1/2 AB 1/2 = AB. Sequential independence for three or more effects was considered in [8] and a more general result was proved. Our next result shows that if A ◦ B is sharp, then A and B are compatible (and hence, sequentially independent). Theorem 9.7.3. [8] For A, B ∈ E(H), if A ◦ B ∈ P(H), then AB = BA. Proof. Assume that A ◦ B ∈ P(H). ® Suppose that A ◦ Bx = x where kxk = 1. We then have BA1/2 x, A1/2 x = 1. By Schwarz’s inequality we have BA1/2 x = A1/2 x and hence, Ax = A ◦ Bx = x. Because x is an eigenvector of A with eigenvalue 1, the same holds for A1/2 . Thus, A1/2 x = x so that BA1/2 x = A ◦ Bx. We conclude that BA1/2 x = A ◦ Bx for all x in the range of A ◦ B. Now suppose that A ◦ Bx = 0. We then have D E kB 1/2 A1/2 xk2 = B 1/2 A1/2 x, B 1/2 A1/2 x = hA ◦ Bx, xi = 0 so that B 1/2 A1/2 x = 0. Hence, BA1/2 x = 0 and it follows that BA1/2 x = A ◦ Bx for all x in the null space of A ◦ B. We conclude that BA1/2 = A ◦ B. Hence, BA1/2 = A ◦ B = (A ◦ B)∗ = A1/2 B so that AB = BA. The last theorem shows why it is important to consider unsharp effects. Even if A and B are sharp, then A ◦ B ∈ / P(H) unless A and B are compatible. Simple examples show that the converse of Theorem 9.7.3 does not hold. However, the converse does hold for sharp effects. Corollary 9.7.4. If A, B ∈ P(H) then A ◦ B ∈ P(H) if and only if AB = BA. It follows from Corollary 9.7.4 that for A, B ∈ P(H) we have A ◦ B = B if and only if AB = BA = B. We now generalize this result to arbitrary effects. Theorem 9.7.5. [8] For A, B ∈ E(H) the following statements are equivalent. (a) A ◦ B = B. (b) B ◦ A = B. (c) AB = BA = B. Proof. It is clear that (c) implies both (a) and (b). It then suffices to show that (a) and (b) each imply (c). If A ◦ B = B we have B 2 A = A1/2 BA1/2 BA = A1/2 B(A1/2 BA1/2 )A1/2 = A1/2 B 2 A1/2 .


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Taking adjoints gives B 2 A = AB 2 . It follows that AB = BA = B. If B ◦ A = B then for every x ∈ H we have E D AB 1/2 x, B 1/2 x = hB ◦ Ax, xi = hBx, xi = kB 1/2 xk2 . If B 1/2 x 6= 0 then

¿ À B 1/2 x B 1/2 x A , = 1. kB 1/2 xk kB 1/2 xk

It follows from Schwarz’s inequality that AB 1/2 x = B 1/2 x. Hence, AB 1/2 = B 1/2 so AB 1/2 = B 1/2 A = B 1/2 . We again conclude that AB = BA = B. Theorem 9.7.5 cannot be strengthened to the case A ◦ B ≤ B. That is A ◦ B ≤ B does not imply AB = BA. For example, in C2 let ∙ ¸ ∙ ¸ 1 30 1 11 , B= ; A= 4 11 4 01 then A ◦ B ≤ B but AB 6= BA. The simplest version of the law of total probability would say that Pρ (B) = Pρ (A)Pρ (B | A) + Pρ (I − A)Pρ (B | I − A) ,

(9.17)

where we interpret I − A as the complement (or negation) of A ∈ E(H). In terms of the sequential product (9.17) can be written as Pρ (B) = Pρ (A ◦ B) + Pρ ((I − A) ◦ B) = Pρ [(A ◦ B + (I − A) ◦ B)] . (9.18) When does (9.18) hold for every ρ ∈ D(H)? Equivalently, when does the following equation hold? B = A ◦ B + (I − A) ◦ B.

(9.19)

This question is also equivalent to finding the fixed points of the L¨ uders map L(B) = A ◦ B + (I − A) ◦ B for B ∈ E(H). Theorem 9.7.6. [5, 8] For A, B ∈ E(H), (9.19) holds if and only if AB = BA. Proof. It is clear that (9.19) holds if AB = BA. Conversely, assume that (9.19) holds and write it as B = A1/2 BA1/2 + (I − A)1/2 B(I − A)1/2 . Multiplying by A1/2 on the left and right, we obtain

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A1/2 BA1/2 = ABA + (I − A)1/2 A1/2 BA1/2 (I − A)1/2 h i = ABA + (I − A)1/2 B − (I − A)1/2 B(I − A)1/2 (I − A)1/2 = ABA − (I − A)B(I − A) + (I − A)1/2 B(I − A)1/2 = ABA − (I − A)B(I − A) + B − A1/2 BA1/2 . Hence, 2A1/2 BA1/2 = ABA − (I − A)B(I − A) + B = AB + BA.

(9.20)

Using the commutator notation [X, Y ] = XY − Y X, (9.20) gives h i A1/2 , [A1/2 , B] = A1/2 (A1/2 B − BA1/2 ) − (A1/2 B − BA1/2 )A1/2 = AB − 2A1/2 BA1/2 + BA = 0.

It follows that for every spectral projection E of A we have h i E, [A1/2 , B] = 0.

By the Jacobi identity i h i h i h E, [A1/2 , B] + B[E, A1/2 ] + A1/2 , [B, E] = 0.

¤ £ We have that A1/2 , [E, B] = 0. As before we obtain [E, [E, B]] = 0. Hence, 0 = E(EB − BE) − (EB − BE)E = EB + BE − 2BE

which we can write as EB = 2EBE − BE. Multiplying on the left by E gives EB = EBE. Hence, EB = (EBE)∗ = BE. It follows that AB = BA. Although the sequential product is always distributive on the right, Theorem 9.7.6 shows that it is not always distributive on the left. That is, (A + B) ◦ C 6= A ◦ C + B ◦ C in general, when A + B ≤ I. Indeed, if AC 6= CA, then by Theorem 9.7.6 we have A ◦ C + (I − A) ◦ C 6= C = [A + (I − A)] ◦ C. One might conjecture that the following generalization of Theorem 9.7.6 holds. If A + B ≤ I and (A + B) ◦ C = A ◦ C + B ◦ C, then CA = AC or CB = BC. However, this conjecture is false. Indeed, suppose that CB 6= BC.


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Nevertheless, we have ¡1 ¢ ¡1 ¢ ¡1 ¢ 1 1 1 2 B + 2 B ◦ C = B ◦ C = 2 B ◦ C + 2 B ◦ C = 2 B ◦ C + 2 B ◦ C.

We close by considering another 9.7.6. Suppose P generalization of Theorem P Ai ∈ E(H), i = 1, . . . , n with Ai = I and that B = Ai ◦ B. Does this imply that BAi = Ai B, i = 1, . . . , n? Notice that the answer is affirmative if AiP ∈ P(H), i = 1, . . . , n. In fact, we only need Ai ∈ P(H), i = 1, . . . , n and PAi ≤ I. In this case, we have Ai Aj = Aj Ai = 0 for i 6= j. Hence, if B = Ai ◦B, then Ai B = BAi = Ai ◦B, i = 1, . . . , n. A proof very similar to that in Theorem 9.5.1 gives an affirmative answer when dim H < ∞ or when B has discrete spectrum with a strictly decreasing sequence of eigenvalues. However, when dim H = ∞ the answer is negative in general [1].

References 1. A. Arias, A. Gheondea, and S. Gudder, “Fixed points of quantum operations,” J. Math. Phys. 43 (2002) 5872—5881. 2. H. Barnum, “Information-disturbance tradeoff in quantum measurement on the uniform ensemble,” Proc. IEEE Intern. Sym. Info. Theor., Washington, D.C., 2001. 3. O Bratteli, P. Jorgensen, A. Kishimoto, and R. Werner, “Pure states on Od ,” J. Operator Theory 43 (2000) 97—143. 4. P. Busch, P. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurements (Springer, Berlin, 1996). 5. P. Busch and J. Singh, “L¨ uders theorem for unsharp quantum effects,” Phys. Lett. A 249 (1998) 10—24. 6. M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Alg. Appl. 10 (1975) 285—290. 7. E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976). 8. S. Gudder and G. Nagy, “Sequential quantum measurements,” J. Math. Phys. 42 (2001) 5212—5222. 9. K. Kraus, States, Effects, and Operations (Springer-Verlag, Berlin, 1983). 10. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). 11. W. Rudin, Functional Analysis (McGraw-Hill, New York, 1991).

Chapter 10

Ekeland Duality as a Paradigm Jean-Paul Penot

Summary. The Ekeland duality scheme is a simple device. We examine its relationships with several classical dualities, such as the Fenchel—Rockafellar duality, the Toland duality, the Wolfe duality, and the quadratic duality. In particular, we show that the Clarke duality is a special case of the Ekeland duality scheme. Key words: Clarke duality, duality, Ekeland duality, Fenchel transform, Legendre function, Legendre transform, nonsmooth analysis

10.1 Introduction Duality is a general tool in mathematics. It consists in transforming a difficult problem into a related one which is more tractable; then, when returning to the initial, or “primal”, problem, some precious information becomes available. Although such a process is of common use in optimization theory and algorithms (see [23, 41, 45] and their references), it pertains to a much larger field. Cramer, Fourier, Laplace, and Radon transforms give testimonies of the power of such a scheme. Even in optimization theory, there is a large spectrum of duality processes: linear programming, convex programming, fractional programming [21], geometric programming, generalized convex programming, quadratic programming [13], semidefinite programming, and so on. It is the purpose of the present chapter to show that several classical duality theories can be cast into a simple general framework. Jean-Paul Penot Laboratoire de mathématiques appliqu´ ees, UMR CNRS 5142, University of Pau, Facult´ e des Sciences, B.P. 1155, 64013 PAU c´ edex, France e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_10, © Springer Science+Business Media, LLC 2009

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A number of physical phenomena can be described by using the minimizers of a suitable potential function; however, it may be sensible to consider that a notion of stationarity is more adapted than minimization or maximization. In a famous paper [14] I. Ekeland introduced a duality scheme that deals with critical points instead of minimizers and takes advantage of the power of the tools of differential topology. In order to extend the reach of his theory we drop the smoothness properties required in [14], following a track indicated in [15]. For such an aim, we make use of elementary notions of nonsmooth analysis recalled in Section 10.4 below. We particularly focus our attention on the convex case for which a close link between the classical Fenchel duality and the Ekeland duality can be obtained thanks to a slight extension of the Brønsted—Rockafellar theorem. But we also consider the concave case, the quadratic case, the Toland duality, and the Clarke duality. The Clarke duality deals with the study of the set of critical points of a function f of the form f (x) :=

1 hAx, xi + g(x) 2

x ∈ X,

where X is a Banach space, A is an self-adjoint operator from X into X ∗ (i.e., hAx, x0 i = hx, Ax0 i for any x, x0 ∈ X) and g : X → R∞ := R ∪ {+∞} is a closed proper convex function. It has been applied to the study of solutions to the Hamilton equation in [5, 7—10, 16—20]. It is the main purpose of the present chapter to endeavor to cast the Clarke duality in the general framework of the Ekeland duality. Such an aim may enhance the interest for this general approach. We also obtain a slight complement to the Clarke duality. On the other hand, we assume that the operator A is continuous (instead of densely defined). This assumption guarantees that the notion of critical point we adopt corresponds to a general and natural concept and is not just an ad hoc specific notion. This new feature is valid for all usual subdifferentials of nonsmooth analysis. This assumption suffices for the application to Hamiltonian systems. In Sections 10.2 and 10.3 we recall the Ekeland duality in the framework of normed vector spaces (n.v.s.). In Section 10.4 we present tools from nonsmooth analysis which enable one to give a rigorous treatment without assuming regularity assumptions. In particular, we introduce a concept of extended Legendre function using methods reminiscent of the notion of limiting subdifferentials (Section 10.5). Such a concept encompasses the case of the Fenchel conjugate of a convex function. Therefore we can apply it to convex duality and show in Section 10.6 that the Fenchel—Rockafellar duality is part of the duality scheme we study. The same is shown for the Toland duality in Section 10.7 and for the Wolfe duality in Section 10.8. The last section is devoted to showing that Clarke duality is a special instance of Ekeland duality. We do not look for completeness but we endeavor to put some light on some significant instances. Duality of integral functionals is considered elsewhere.

10 Ekeland Duality as a Paradigm

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Duality in the calculus of variations using the Ekeland’s scheme is performed in [14] and [15]. Because, as mentioned above, many phenomena in physics and mechanics can be modeled by using critical point theory rather than minimization, we believe that the extensive approach by D. Gao and his co-authors (see [22—29] and their references) deserves some more attention and should be combined with the present contribution. In the sequel P stands for the set of positive real numbers, B(0, r) is the open ball with center 0 and radius r, and SX := {u ∈ X : kuk = 1} is the unit sphere in a normed vector space.

10.2 Preliminaries: The Ekeland—Legendre Transform The Ekeland duality deals with the search of critical points and critical values of functions or multifunctions. It can be cast in a general framework in which there is no linear structure (see [44]), but here we remain in the framework of normed vector spaces (n.v.s.) in duality. Definition 10.1. Given two n.v.s. X, X 0 and a subset J of X × X 0 × R, a pair (x, r) is called a critical pair of J if (x, 0X 0 , r) ∈ J. A point x of X is called a critical point of J if there exists some r ∈ R such that (x, r) is a critical pair of J. A real number r is called a critical value of J if there exists some x ∈ X such that (x, r) is a critical pair of J. The extremization of J consists in the determination of the set ext J of critical pairs of J. When J is a generalized 1-jet in the sense that the projection G of J on X × R is the graph of a function j : X0 → R defined on some subset X0 of X, the extremization of j is reduced to the search of critical points of J. Note that J is a generalized 1-jet if and only if one has the implication (x1 , x01 , r1 ) ∈ J,

(x2 , x02 , r2 ) ∈ J,

x1 = x2 =⇒ r1 = r2 .

Example 10.1. In the classical case X 0 is the topological dual space X ∗ of X and J is the 1-jet J 1 j of a differentiable function j : X0 → R, where X0 is an open subset of X, defined by J 1 j := {(x, Dj(x), j(x)) : x ∈ X0 }, where Dj(x) is the derivative of j at x. Then we recover the usual notion. One may also suppose as in [14] that X0 is a differentiable submanifold in X and replace Dj(x) by djx , the restriction to the tangent space to X0 at x of the 1-form dj. The fact that J may be different from a 1-jet gives a great versatility to the duality which is exposed.

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Example 10.2. Given a convex function j : X → R∞ := R ∪ {+∞}, let X 0 be the topological dual space X ∗ of X and let J be the subjet of j, defined by J := {(x, x∗ , j(x)) : x ∈ dom j, x∗ ∈ ∂j(x)},

where dom j := j −1 (R) and ∂j(x) ⊂ X ∗ is the Fenchel—Moreau subdifferential of j at x given by x∗ ∈ ∂j(x) ⇔ j(·) ≥ x∗ (·) + j(x) − x∗ (x). Then the extremization of J coincides with the minimization of j.

In view of its importance for the sequel, let us anticipate Section 10.4 by presenting the next example. Example 10.3. Let J be the subjet J ∂ j of a function j : X → R∞ := R ∪ {∞} associated with some subdifferential ∂: J ∂ j := {(x, x0 , r) ∈ X × X 0 × R : x0 ∈ ∂j(x), r = j(x)}. In such a case, ext J is the set of pairs (x, r) such that 0X 0 ∈ ∂j(x), r = j(x). We make clear what we mean by “subdifferential” in Section 10.4. For the moment we may take for ∂j either the proximal subdifferential ∂ P j of j, given by x∗ ∈ ∂ P j(x) iff 2

j(x + u) ≥ x∗ (u) + j(x) − c kuk ,

∃c, r ∈ P : ∀u ∈ B(0, r)

or the Fréchet (or firm) subdifferential ∂ F j of j given by x∗ ∈ ∂ F f (x) iff ∃α ∈ A : ∀u ∈ X

j(x + u) ≥ x∗ (u) + j(x) − α(kuk) kuk ,

where A is the set of functions α : R+ → R+ ∪{+∞} satisfying limr→0 α(r) = 0, or the Dini—Hadamard (or directional) subdifferential ∂ D j of j given by x∗ ∈ ∂ D f (x) iff ∀u ∈ SX , ∃α ∈ A : ∀(v, t) ∈ X×R+ j(x+tv) ≥ x∗ (tv)+j(x)−α(ku − vk+t)t, or the Clarke—Rockafellar subdifferential given by x∗ ∈ ∂ C j(x) iff ∃α ∈ A : ∀(x0 , v, t) ∈ X 2 × R+

j(x0 + tv) ≥ x∗ (tv) + j(x0 ) − α(s)t,

with s := ku − vk+kx0 − xk+t (in the case where f is continuous). Of course, in the preceding definitions we assume j is finite at x and we take the empty set otherwise. We can generalize the preceding cases by considering other subdifferentials appropriate for nonconvex functions (here we have chosen the most usual subdifferentials among classical ones).


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Example 10.4. Let j : X → R be a concave function and let J be the subjet J ∂ j of j for one of the first three preceding subdifferentials. Then the extremization of J leads to the maximization of j. In fact, if x∗ ∈ ∂j(x), then for all u ∈ X one has j 0 (x, u) :=

1 (j(x + tv) − j(x)) ≥ x∗ (u), (t,v)→(0+ ,u) t lim

so that j is Hadamard differentiable at x, with derivative x∗ . Thus −x∗ ∈ ∂(−j)(x) and if x∗ = 0 we get that x is a maximizer of j. If x∗ ∈ ∂ C j(x) and j is continuous, we also have −x∗ ∈ ∂ C (−j)(x) = ∂(−j)(x) because j is locally Lipschitzian. Example 10.5. Given a subdifferential ∂ and a function j : X → R∞ , let J := {(x, x0 , r) ∈ X ×X 0 ×R : x0 ∈ Υ j(x) := ∂j(x)∪(−∂(−j)(x)) , r = j(x)}. This choice is justified by the case where j is concave. In such a case, a pair (x, r) is critical if and only if x is a maximizer of j and r = max j(X): the condition is sufficient because for any maximizer x of j one has 0 ∈ ∂(−j)(x); we have seen that it is necessary when 0 ∈ ∂j(x) and it is obviously necessary when 0 ∈ −∂(−j)(x) because −j is convex. Example 10.6. Let j be a d.c. function, that is, a function of the form j := g − h, where g and h are convex functions on some convex subset of X. Let J := {(x, x0 , r) ∈ X × X 0 × R : x0 ∈ ∂g(x) ¯ ∂h(x), r = j(x)},

where, for two subsets C, D of X 0 , C ¯ D denotes the set of x0 ∈ X 0 such that D + x0 ⊂ C. Some sufficient conditions ensuring that ∂g(x) ¯ ∂h(x) coincides with the Fréchet subdifferential of j are known [1]; but in general J is different from J F j.

Example 10.7. Let (S, S, σ) be a measured space, let E be a Banach space, and let : S × E → R be a measurable integrand, with which is associated the integral functional j given by Z j(x) := (s, x(s))dσ(s) x ∈ X, S

where X is some normed vector space of (classes of) measurable functions from S to E; for instance X := Lp (S, E) for some p ∈ [1, +∞[. Then, if X 0 is a space of measurable functions from S to the dual E ∗ of E (for instance X 0 := Lq (S, E ∗ ), with q := (1 − p−1 )−1 ) one can take J := {(x, x0 , r) ∈ X × X 0 × R : x0 (s) ∈ ∂ s (x(s)) a.e. s ∈ S, r = j(x)}, where s := (s, ·). One can give conditions ensuring that J is exactly the subjet of j; but in general that is not the case.

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Let us present another example of a different kind bearing on mathematical programming. Example 10.8. Let X and Z be n.v.s. with dual spaces X ∗ and Z ∗ , respectively. Given a closed convex cone C in Z and differentiable maps f : X → R, g : X → Z, let J := {(x, f 0 (x) + z ∗ ◦ g 0 (x), f (x)) : z ∗ ∈ C 0 , hz ∗ , g(x)i = 0}, where C 0 := {z ∗ ∈ Z ∗ : hz ∗ , zi ≤ 0 ∀z ∈ C} is the polar cone of C. This choice is clearly dictated by the Karush—Kuhn—Tucker optimality conditions. But, as is well known, a solution of the mathematical programming problem (M)

minimize f (x) subject to g(x) ∈ C

is a critical point for J only when some qualification condition is satisfied. The approach of Ekeland to duality [14, 15] can be extended to the case of an arbitrary coupling (see [44]). Here we limit our study to bilinear couplings. The normed vector space X appearing in the following definition is usually a space of parameters and X 0 is usually its topological dual space, but other cases may be considered. Definition 10.2. Given two normed vector spaces X, X 0 paired by a bilinear coupling function c : X × X 0 → R, the Ekeland (or Legendre) map is the mapping E : X × X 0 × R → X 0 × X × R given by E(x, x0 , r) := (x0 , x, c(x, x0 ) − r). Clearly, E is a kind of involution: denoting by E 0 the mapping E 0 : X 0 × X × R → X × X 0 × R given by E 0 (x0 , x, r) := (x, x0 , c(x, x0 ) − r), one has E ◦ E 0 = I, E 0 ◦ E = I, so that E −1 = E 0 and E 0 has a similar form. In particular, when X 0 = X, one has E 0 = E, and E is a true involution. We show that under appropriate assumptions, the transform E induces a kind of conjugacy between functions on X and on X 0 . It can also be applied to multifunctions. Definition 10.3. Given paired n.v.s. X and X 0 , the Ekeland transform J E of a subset J of X × X 0 × R is the image of J by E: J E := E(J).

10.3 The Ekeland Duality Scheme In the present chapter the decision space X and the parameter space W play a symmetric role; it is not the case in [44] where X is supposed to be an arbitrary set. We assume X and W are n.v.s. paired with n.v.s. W 0 and X 0 , respectively, by couplings denoted by cW , cX , or just h·, ·i if there is no risk of confusion. Then we put Z := W × X in duality with X 0 × W 0 by the means


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of the coupling c given by c((w, x), (x0 , w0 )) = cW (w, w0 ) + cX (x, x0 ).

(10.1)

Such an unorthodox coupling is convenient in the sequel. The following definition is reminiscent of the notion of perturbation which is one of the two main approaches to duality in convex analysis. However, it is taken in a more restrictive sense when J is the subjet of some convex function, unless the convex function is continuous. Definition 10.4. Given two pairs (W, W 0 ), (X, X 0 ) of n.v.s. in duality, and a subset J ⊂ X × X 0 × R, a subset P of W × X × X 0 × W 0 × R is said to be a hyperperturbation of J if J = {(x, x0 , r) ∈ X × X 0 × R : ∃w0 ∈ W 0 , (0W , x, x0 , w0 , r) ∈ P }. A subset P of W ×X ×X 0 ×W 0 ×R is said to be a critical perturbation of J if (x, 0X 0 , r) ∈ J ⇔ ∃w0 ∈ W 0 , (0W , x, 0X 0 , w0 , r) ∈ P. In other terms, P is a hyperperturbation of J if J coincides with the domain of the slice P0 : X × X 0 × R ⇒ W 0 of P given by P0 (x, x0 , r) := {w0 ∈ W 0 : (0W , x, x0 , w0 , r) ∈ P }. In order to study the extremization problem (P)

find (x, r) ∈ X × R such that (x, 0X 0 , r) ∈ J,

given a critical perturbation P of J and a coupling c : W ×W 0 → R, following Ekeland [14, 15] one can introduce the transform P 0 := E(P ) ⊂ X 0 × W 0 × W × X × R of P given by P 0 := {(x0 , w0 , w, x, hw0 , wi + hx0 , xi − r) : (w, x, x0 , w0 , r) ∈ P }. The domain J 0 = {(w0 , w, r0 ) ∈ W 0 × W × R : ∃x ∈ X, (0X 0 , w0 , w, x, r0 ) ∈ P 0 } of the slice P00 : W 0 × W × R ⇒ X of P 0 given by P00 (w0 , w, r0 ) := {x ∈ X : (0X 0 , w0 , w, x, r0 ) ∈ P 0 } yields the extremization problem (P 0 )

find (w0 , r0 ) ∈ W 0 × R such that (w0 , 0W , r0 ) ∈ J 0

called the adjoint problem. Denoting by ext J the solution set of (P) (i.e., the set of (x, r) ∈ X × R such that (x, 0X 0 , r) ∈ J) and by ext J 0 the solution set of (P 0 ), one has the following result.

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Theorem 10.1. Let J be a subset of X ×X 0 ×R. For any critical perturbation P of J, the set P 0 := E(P ) defined as above is a hyperperturbation of J 0 , hence is a critical perturbation of J 0 . Moreover, the problems (P) and (P 0 ) are in duality in the following sense. (a) If (w0 , r0 ) ∈ ext J 0 , then P00 (w0 , 0W , r0 ) is nonempty and for any x ∈ one has (x, −r0 ) ∈ ext J. (b) If (x, r) ∈ ext J, then P0 (x, 0X 0 , r) is nonempty and for any w0 ∈ P0 (x, 0X 0 , r) one has (w0 , −r) ∈ ext J 0 . (c) The set of critical values of (P) is the opposite of the set of critical values of (P 0 ).

P00 (w0 , 0W , r0 )

Proof. The first assertion is an immediate consequence of the definition of P 0 and J 0 : a pair (w0 , r0 ) ∈ W 0 × R is in ext J 0 if and only if there exists some x ∈ X such that (0X 0 , w0 , 0W , x, r0 ) ∈ P 0 ; that is, x ∈ P00 (w0 , 0W , r0 ). For any such x one has (0W , x, 0X 0 , w0 , −r0 ) ∈ P , hence (x, 0X 0 , −r0 ) ∈ J or (x, −r0 ) ∈ ext J. Assertion (b) similarly results from the implications (x, r) ∈ ext J ⇔ (x, 0X 0 , r) ∈ J ⇔ ∃w0 ∈ W 0 : (0W , x, 0X 0 , w0 , r) ∈ P ⇔ ∃w0 ∈ W 0 : (0X 0 , w0 , 0W , x, −r) ∈ P 0 so that for any w0 ∈ P0 (x, 0X 0 , r) one has x ∈ P00 (w0 , 0W , −r); that is, (w0 , −r) ∈ ext J 0 . Assertion (c) is part of the preceding analysis. u t The problem (P ∗ )

find (w0 , r) ∈ W 0 × R such that (w0 , 0W , −r) ∈ J 0

can be called the dual problem of (P). The preceding result is akin to [15, Proposition 3] which deals with the enlarged problem (E 0 )

find (w0 , x, r0 ) ∈ W 0 × X × R such that (0X 0 , w0 , 0W , x, r0 ) ∈ P 0 .

It clearly corresponds to the problem (E)

find (x, w0 , r) ∈ X × W 0 × R such that (0W , x, 0X 0 , w0 , r) ∈ P

via the relation r0 = −r. [15, Proposition 3] is subsumed by the following statement. Each of its assertions implies that (x, r) is a solution to (P) and (w0 , r0 ) is a solution to (P 0 ) for r = −r0 . Proposition 10.1. For an element (w0 , x, r0 ) of W 0 × X × R the following assertions are equivalent. (a) (w0 , x, r0 ) is a solution to (E 0 ). (b) (x, r) with r := −r0 is a solution to (P) and w0 ∈ P0 (x, 0X 0 , −r0 ). (c) (w0 , r0 ) is a solution to (P 0 ) and x ∈ P00 (w0 , 0W , r0 ).


Proof. Each assertion is equivalent to (0W , x, 0X 0 , w0 , −r) ∈ P .

357

t u

We notice that applying to P 0 the same process, we get an enlarged problem (E 00 ) which coincides with (E). Thus, as for (P) and (P 0 ) we have an appealing symmetry.

10.4 Tools from Nonsmooth Analysis A case of special interest arises when the perturbation set P is the subjet of some function p : W × X → R. Although its Ekeland transform is not necessarily a subjet, in some cases one can associate a function with it. In such a case, the dual problem becomes close to the classical dual problem, as we show in the following sections. In order to deal with such a nice situation we need to give precise definitions. Let us first make clear what we mean by “subdifferential.” Here, given a n.v.s. X with dual X 0 = X ∗ , a set F(X) ⊂ RX ∞ of functions on X with values in R∞ , a subdifferential is a map ∂ : F(X) × X → P(X 0 ) with values in the space of subsets of X 0 which associates with a pair (f, x) ∈ RX ∞ × X a subset ∂f (x) of X 0 which is empty if x is not in the domain dom f := {x ∈ X : f (x) ∈ R} of f and such that (M) If x is a minimizer of f , then 0X 0 ∈ ∂f (x). Thus, minimizers are critical points. We do not look for a list of axioms, although such lists exist ([4, 30—32, 39] and others). However, we may require some other conditions such as the following ones in which X, Y , Z are n.v.s. and L(X, Y ) denotes the space of linear continuous maps from X into Y : (F) If f is convex, ∂f coincides with the Fenchel—Moreau subdifferential: ∂f (x) := {x∗ ∈ X ∗ : f (·) ≥ x∗ (·) − x∗ (x) + f (x)}. (T) If f := g + h, where h is continuously differentiable at x, then ∂f (x) = ∂g(x) + Dh(x). (T0 ) If f is continuously differentiable at x, then ∂f (x) = {Df (x)}. (C) If f := g ◦ , where ∈ L(X, Y ) and g ∈ RY∞ , then ∂g( (x))◦ ⊂ ∂f (x). (C0 ) If f := g◦ , with ∈ L(X, Y ) open, g ∈ RY∞ , then ∂g( (x))◦ ⊂ ∂f (x). (O) If f := g ◦ , where ∈ L(X, Y ) is open and g : Y → R is locally Lipschitzian, then ∂f (x) ⊂ ∂g( (x)) ◦ . (P) If f := g ◦ pY , where pY : Y × Z → Y is the canonical projection and g ∈ RY∞ , then ∂f (y, z) = ∂g(y) ◦ pY . (D) If f := g ◦ , where ∈ L(X, Y ) is an isomorphism and g ∈ RW ∞ , then ∂f (x) = ∂g( (x)) ◦ . Clearly (T0 ) is a special case of the translation property (T) and (P) is a special case of the conjunction of the composition properties (C) and (O).

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Condition (D) which can be considered as a very special case of (P) is satisfied by all usual subdifferentials. Other relationships are described in the following statement. Proposition 10.2. (a) If ∂ is either the Fréchet subdifferential or the Hadamard subdifferential then conditions (F), (T), (C), and (O) are satisfied. (b) If ∂ is either the Clarke subdifferential [6] or the moderate subdifferential [33] then conditions (F), (T), (C0 ), and (O) are satisfied. Proof. (a) The coincidence with the Fenchel subdifferential (F), the translation property (T), the composition properties (C) and (O) are easy to check. Let us prove the two latest ones. Given x ∈ X, ∈ L(X, Y ), and y ∗ ∈ ∂ D g(y), with y := (x), we observe that for every u ∈ X we have f 0 (x, u) ≥ g 0 (y, (u)) ≥ hy ∗ , (u)i. Thus y ∗ ◦ ∈ ∂ D f (x). If y ∗ ∈ ∂ F g(y), one can find some function β : Y → R such that limv→0 β(v) = 0 and g(y + v) − g(y) − hy ∗ , vi ≥ −β(v) kvk for v in a neighborhood V of 0 in Y . Then, for u ∈ U :=

−1

(V ) one has

f (x + u) − f (x) − hy ∗ ◦ , ui ≥ −β( (u)) k k kuk , so that y ∗ ◦ ∈ ∂ F f (x). Now suppose is open. Because BY ⊂ (cBX ), for some c > 0, where BX , BY are the closed unit balls of X and Y , respectively, for every unit vector v ∈ Y we can pick some u ∈ cBX such that (u) = v. By homogeneity, we obtain a map h : Y → X such that (h(v)) = v and kh(v)k ≤ c kvk for all v ∈ Y . Let x∗ ∈ ∂ D f (x). Because g is locally Lipschitzian, for all u ∈ X we have hx∗ , ui ≤ f 0 (x, u) = g 0 (y, (u)) and g 0 (y, 0) = 0. Thus, hx∗ , ui = 0 for all u in the kernel N of . Because is open, it follows that there exists some y ∗ ∈ Y ∗ such that x∗ = y ∗ ◦ . From the surjectivity of and the relation hy ∗ ◦ , ui ≤ g 0 (y, (u)) for all u ∈ X we conclude that y ∗ ∈ ∂ D g(y). Now let us suppose x∗ ∈ ∂ F f (x). By what precedes we obtain that there exists some y ∗ ∈ ∂ D g(y) such that x∗ = y ∗ ◦ . Let α : X → R and r > 0 be such that limu→0 α(u) = 0 and f (x + u) − f (x) − hx∗ , ui ≥ −α(u) kuk for u ∈ rBX . Let h : Y → X be the map constructed above, and let s := c−1 r. Because for all v ∈ sBY we have h(v) ∈ rBX , we get, as hy ∗ , vi = hy ∗ , (h(v))i = hx∗ , h(v)i, g(y) = f (x), and f (x + h(v)) = g( (x) + (h(v))) = g(y + v), g(y + v) − g(y) − hy ∗ , vi ≥ −β(v) kvk with β(v) := cα(h(v)) → 0 as v → 0. Thus y ∗ ∈ ∂ F g(y).


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(b) Again, the assertions concerning (F) and (T) are classical and elementary. For the Clarke subdifferential, the assertions concerning (C0 ) and (O) are particular cases of [6, Theorem 2.3.10]. The case of the moderate subdifferential is similar. t u Let us insist on the fact that extremization problems are not limited to the examples mentioned in the previous sections. In particular, one may take for J some subset of the closure of a subjet with respect to some topology (or convergence) on X × X 0 × R. Another case of interest appears when X is a n.v.s. and J is the hypergraph of a multifunction M : X ⇒ R associated with a notion of normal cone: H(M ) := {(x, x∗ , r) ∈ X × X ∗ × R : (x∗ , −1) ∈ N (G(M ), (x, r)), r ∈ M (x)}, where G(M ) is the graph of M and N (G(M ), (x, r)) denotes the normal cone to G(M ) at (x, r). The normal cone N (S, s) at s to a subset S of a n.v.s. X can be defined in different ways. Some axiomatic approach can be adopted as in [40]. When one disposes of a subdifferential ∂ on the set of Lipschitzian functions on X one may set N (S, s) := R+ ∂dS (s), where dS is the distance function to S: dS (x) := inf{d(x, y) : y ∈ S}. When the subdifferential ∂ is defined over the set S(X) of lower semicontinuous functions on X, one can also define N (S, s) by N (S, s) := ∂ιS (s), where ιS is the indicator function of S given by ιS (x) = 0 for x ∈ S, +∞ else. Introducing the coderivative D∗ M (x, r) of M at (x, r) ∈ G(M ) by D∗ M (x, r) := {x∗ ∈ X ∗ : (x∗ , −1) ∈ N (G(M ), (x, r))}, we see that H(M ) is the set of (x, x∗ , r) ∈ X × X ∗ × R such that x∗ ∈ D∗ M (x, r). In particular, if M is the epigraph multifunction of a function f , H(M ) coincides with J ∂ f whenever x∗ ∈ ∂f (x) if and only if (x∗ , −1) ∈ N (epi(f ), (x, f (x))). When M is a hypergraph, E(M ) is not necessarily a hypergraph. When M is the subjet J ∂ f associated with a function f on X and a subdifferential ∂, the set E(M ) is not necessarily the subjet of some function on X 0 . It is of interest to introduce a notion that implies part of such a requirement. This is the aim of the next section.

10.5 Ekeland and Legendre Functions We first delineate a class of functions for which a conjugate function can be defined. Definition 10.5. [42] Given a pairing c between the n.v.s. X and X 0 and a subdifferential ∂ : F(X) × X → P(X 0 ), a function f ∈ F(X) is an Ekeland function with respect to ∂, in short an ∂-Ekeland function, or just an Ekeland

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function if there is no risk of confusion, if for any x1 , x2 ∈ X, x0 ∈ X 0 satisfying x0 ∈ ∂f (x1 ) ∩ ∂f (x2 ) one has c(x1 , x0 ) − f (x1 ) = c(x2 , x0 ) − f (x2 ). Then, the Ekeland transform of f is the function f E : X 0 → R∞ given by f E (x0 ) := c(x, x0 ) − f (x) for x ∈ (∂f )−1 (x0 ) for x0 ∈ ∂f (X), f E (x0 ) = +∞ for x0 ∈ X 0 \∂f (X). Thus, the graph of f E is the projection on X 0 × R of E(J ∂ f ). Example 10.9. Any convex function (on some n.v.s.) is an Ekeland function for any subdifferential satisfying condition (F). In fact, for any given x0 ∈ X 0 , every x ∈ (∂f )−1 (x0 ) is a maximizer of the function c(·, x0 ) − f (·) so that the value of this function at x is independent of the choice of x. Example 10.10. Any concave function on some n.v.s. X is an Ekeland function for the Fréchet and the Dini—Hadamard subdifferentials. In fact, for any x1 , x2 ∈ X, x∗ ∈ X ∗ satisfying x∗ ∈ ∂f (x1 ) ∩ ∂f (x2 ) one has hx∗ , x1 i − f (x1 ) = hx∗ , x2 i − f (x2 ) because in such a case x∗ is the Hadamard derivative of f at xi (i = 1, 2), hence hx∗ , xi i − f (xi ) = min{hx∗ , xi − f (x) : x ∈ X}. Then f E is the restriction to f 0 (X) of the concave conjugate f∗ of f . Similar assertions hold when f is continuous. Example 10.11. Let f be a linear-quadratic function on X; that is, f (x) := 1 0 2 hAx, xi−hb, xi+c for some continuous symmetric linear map A : X → X := ∗ 0 X , b ∈ X , c ∈ R. Let ∂ be a subdifferential satisfying condition (T0 ), such as the Clarke, the Fréchet, the Hadamard or the moderate subdifferential. Then f is an Ekeland function. In fact, given x0 ∈ X 0 , x1 , x2 ∈ X such that f 0 (xi ) = x0 one has 1 1 hx0 , xi i − f (xi ) = hAxi − b, xi i − hAxi , xi i + hb, xi i − c = hAxi , xi i − c 2 2 and hAx1 , x1 i − hAx2 , x2 i = hA(x1 − x2 ), x1 i + hAx2 , x1 − x2 i = 0 because A is symmetric and Ax1 = x0 + b = Ax2 , so that A(x1 − x2 ) = 0. Thus, for x0 ∈ A(X) − b, we can write f E (x0 ) = 12 hx0 + b, A−1 (x0 + b)i − c, even if A is noninvertible. Example 10.12. Let f : X → R be a partially quadratic function in the sense that there exist a decomposition X = X1 ⊕ X2 as a topological direct sum, an isomorphism A : X1 → X10 , where X10 := X2⊥ := {x0 ∈ X 0 := X ∗ : x0 | X2 = 0}, b ∈ X10 , c ∈ R such that f (x) := 12 hAx, xi − hb, xi + c for x ∈ X1 , f (x) = +∞ for x ∈ X\X1 . Let ∂ be the Clarke, the Fréchet, the Dini—Hadamard, or the moderate subdifferential. Then, for x ∈ X1 one has ∂f (x) = Ax + X20 , where X20 := X1⊥ . Then, as in the preceding example, one −1 sees that for any x0 ∈ X 0 , x ∈ (∂f ) (x0 ) the value of hx0 , xi − f (x) does not −1 depend on the choice of x in (∂f ) (x0 ). Thus f is an Ekeland function. u t


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The following definition stems from the wish to get a concept which is more symmetric than the notion of Ekeland function. It is also motivated by the convex (and the concave) case in which the domain of f E is the image of ∂f (respectively, f 0 ) which is not necessarily convex, whereas a natural extension of f E is the Fenchel conjugate whose domain is convex and which enjoys nice properties (lower semicontinuity, local Lipschitz property on the interior of its domain, etc.). Definition 10.6. Let X and X 0 be n.v.s. paired by a coupling function c : X × X 0 → R. A l.s.c. function f : X → R∞ is said to be a (generalized) Legendre function for a subdifferential ∂ if there exists a l.s.c. function f L : X 0 → R∞ such that

(a) f and f L are Ekeland functions and f L | ∂f (X) = f E | ∂f (X). (b) For any x ∈ dom f there is a sequence (xn , x0n , rn )n in J ∂ f such that (xn , hxn − x, x0n i, rn ) → (x, 0, f (x)). (b0 ) For any x0 ∈ dom f L there is a sequence (x0n , xn , sn )n in J ∂ f L such that (x0n , hxn , x0n − x0 i, sn ) → (x0 , 0, f L (x0 )). (c) The relations x ∈ X, x0 ∈ ∂f (x) are equivalent to x0 ∈ X 0 , x ∈ ∂f L (x0 ). Condition (b) (resp., (b0 )) ensures that f (resp., f L ) is determined by its restriction to dom ∂f (resp., dom ∂f L ). In fact, for any x ∈ dom f one has f (x) =

lim inf

u(∈dom ∂f )→x

f (u)

because f (x) ≤ lim inf u→x f (u) and (b) implies f (x) = limn f (xn ) for some sequence (xn ) → x in dom ∂f . Moreover, conditions (a) and (b0 ) imply that f L is determined by f . Condition (b) can be simplified when ∂f is locally bounded on the domain of f . In that case, condition (b) is equivalent to the simpler condition (b0 ) For any x ∈ dom f there exists a sequence (xn )n in dom ∂f such that (xn , f (xn )) → (x, f (x)). Example 10.13. Any classical Legendre function is a (generalized) Legendre function. We say that a function f : U → R on an open subset U of a n.v.s. X is a classical Legendre function if it is of class C 2 on U and if its derivative Df is a diffeomorphism from U onto an open subset U 0 of X ∗ . In fact, one can show that it suffices that f be of class C 1 and that its derivative Df be a locally Lipschitzian homeomorphism from U onto an open subset U 0 of X ∗ whose inverse is also locally Lipschitzian. See [42, 43] for such refinements. In particular, let f be the linear-quadratic function on X given by f (x) := (1/2) hAx, xi−hb, xi+c for some symmetric isomorphism A : X → X 0 := X ∗ , b ∈ X 0 , c ∈ R. Then f is a classical Legendre function because Df : x 7→ Ax − b is a diffeomorphism. Example 10.14. A variant is the notion of Legendre—Hadamard function. A function f : U → R on an open subset U of a normed vector space X is a

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Legendre—Hadamard function if it is Hadamard differentiable, if its derivative Df : U → X 0 := X ∗ is a bijection onto an open subset U 0 of X 0 which is continuous when X is endowed with its strong topology and X 0 is endowed with the topology of uniform convergence on compact subsets, its inverse h satisfying a similar continuity property and the Ekeland transform f E of f given by f E (x0 ) := hh(x0 ), x0 i − f (h(x0 )) x0 ∈ U 0

being Hadamard differentiable with derivative h. Then f and f E are of class T 1 in the sense that they are Hadamard differentiable and the functions df : U × X → R and df E : U 0 × X 0 → R given by df (u, x) := Df (u)(x), df E (u0 , x0 ) := Df E (u0 )(x0 ) are continuous (see [37]). Then f is a generalized Legendre function for the Dini—Hadamard subdifferential. In fact, if x0 ∈ ∂f (x) for some x ∈ U , one has x0 = Df (x), hence x = h(x0 ), f E (x0 ) = hx, x0 i − f (x) and ∂f E (x0 ) = {h(x0 )} = {x}, so that conditions (a) and (c) of the preceding definition are satisfied. Conditions (b) and (b0 ) are immediate and in fact, for any x ∈ U and any sequence (xn ) → x one has hxn − x, f 0 (xn )i → 0 and a similar property for f E by the assumed continuity property. Let us give a criterion which has some analogy with the one we gave in the preceding example. Now, the differentiability assumption on f is weaker, but the local Lipschitz condition on the inverse h of Df is changed into the assumption that for any x0 ∈ U 0 the map h is directionally compact at x0 in the following sense: for any v 0 ∈ X 0 and any sequences (vn0 ) → v 0 , (tn ) → 0+ 0 0 0 the sequence (t−1 n (h(x + tn vn ) − h(x )) is contained in a compact set. Such an assumption is satisfied when h is Hadamard differentiable at any x0 or when X is finite-dimensional and h is locally Lipschitzian. Proposition 10.3. Suppose f is of class T 1 and its derivative Df is a bijection from U onto an open subset U 0 whose inverse h is directionally compact at every point of U 0 . Suppose the mappings df : (x, v) 7→ Df (x)(v) and (x0 , v0 ) 7→ h(x0 )(v0 ) are continuous from U × X into R and from U 0 × X 0 into R, respectively. Then f is a Legendre—Hadamard function. Proof. It suffices to prove that f E is Hadamard differentiable at any x0 ∈ U 0 , with derivative h(x0 ). Let v0 ∈ X 0 and let (vn0 ) → v 0 , (tn ) → 0+ . Let 0 0 0 0 us set vn := t−1 n (h(x + tn vn ) − h(x )), x := h(x ). By our assumption of directional compactness, (vn ) is contained in a compact subset of X, so that 0 αn := t−1 n (f (x + tn vn ) − f (x) − tn Df (x)(vn )) has limit 0, x = Df (x) and f E (x0 + tn vn0 ) − f E (x0 ) = hx0 + tn vn0 , h(x0 + tn vn0 )i − f (h(x0 + tn vn0 )) − hx0 , h(x0 )i + f (h(x0 )) = hx0 + tn vn0 , x + tn vn i − f (x + tn vn ) − hx0 , xi + f (x) = tn hvn0 , xi + tn hx0 , vn i + t2n hvn0 , vn i − tn Df (x)(vn ) − tn αn = tn hvn0 , xi + tn βn


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with βn := tn hvn0 , vn i − αn → 0. This shows that f E is Hadamard differentiable at x0 , with derivative x := h(x0 ). t u Example 10.15. Any l.s.c. proper convex function f is a (generalized) Legendre function. In fact, a slight strengthening [38, Proposition 1.1] of the Brønsted—Rockafellar theorem ensures that for any x ∈ dom f there exists a sequence (xn , x∗n ) in the graph of ∂f such that (hxn − x, x∗n i) → 0 and (f (xn )) → f (x). The same is valid for the Fenchel conjugate function f E = f ∗ . Moreover, as is well known, condition (c) holds in such a case. Example 10.16. Let f : X → R ∪ {−∞} be a concave function such that U := dom(−f ) and U 0 := dom((−f )∗ ) are open and f and its concave conjugate f∗ are differentiable on U and U 0 , respectively; here f∗ is given by f∗ (x0 ) = inf x∈X (hx0 , xi − f (x)) = −(−f )∗ (−x0 ) and differentiability is taken in the sense of Fréchet (resp., Hadamard) when one takes the Fréchet (resp., Hadamard) subdifferential. Then f is a generalized Legendre function for this subdifferential ∂. In fact, x0 ∈ ∂f (x) if and only if f is Fréchet (resp., Hadamard) differentiable at x and x0 = f 0 (x). Then, for g := −f , one has −x0 = g 0 (x), hence x ∈ ∂g ∗ (−x0 ). Because f∗ is supposed to be differentiable, 0 0 g ∗ is also differentiable and x = (g ∗ ) (−x0 ) = (f∗ ) (x0 ) ∈ ∂f∗ (x0 ). Moreover, E 0 0 0 one has f (x ) = hx , xi − f (x) = f∗ (x ). Condition (b) is satisfied because for any x ∈ U one can take (xn , x0n , rn ) = (x, f 0 (x), f (x)). Because the roles of f and f∗ are symmetric, we see that f is a generalized Legendre function. Remark. Let U be an open convex subset of an Asplund space X with the Radon—Nikodym property. Let f be a continuous concave function on U such that its concave conjugate f∗ is finite and continuous on an open convex subset U 0 of X 0 and −∞ on X 0 \U 0 . Let ∂ be either the Fréchet or the Hadamard subdifferential and let f L := f∗ . As in the preceding example we see that for any x0 ∈ ∂f (X) one has f L (x0 ) = f E (x0 ). By definition of an Asplund space, f is Fréchet differentiable on a dense subset D of U . Because it is also locally Lipschitzian, its derivative is locally bounded on D. Thus, if x ∈ U and if (xn ) is a sequence of D with limit x, then (hf 0 (xn ), xn − xi) → 0. Now, because f∗ is defined on an open convex subset and is continuous and upper semicontinuous for the weak∗ topology, it is also Fréchet differentiable on a dense subset of its domain by a result of Collier [11] and by a similar argument, we see that condition (b0 ) is satisfied. However, condition (c) is not necessarily satisfied. For example, let X be a Hilbert space, and let f be 2 given by f (x) := − max(kxk , kxk ). Then f∗ (x0 ) = −1 − kx0 k for x0 ∈ 2B 0 , 2 0 where B is the closed unit ball of X 0 and f∗ (x0 ) = − 14 kx0 k for x0 ∈ X 0 \2B 0 . 0 0 Let u be a unit vector in X and let u ∈ X be such that hu0 , ui = −2, ku0 k = 2; then we have u ∈ ∂ F f∗ (u0 ) but u0 ∈ / ∂ F f (u) because f is not Fréchet differentiable at u. Example 10.17. Let ∂ be a subdifferential such that ∂(−f )(x) = −∂f (x) when f is locally Lipschitzian around x. For instance ∂ may be the Clarke

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subdifferential [6], the moderate subdifferential [33], or be given as Υ f (x) := ∂ F f (x) ∪ (−∂ F (−f )(x)) or ∂ D f (x) ∪ (−∂ D (−f )(x)). Let f be a concave function such that −f and −f∗ have open domains and are continuous on their domains. Then f is a generalized Legendre function. In fact, using the notation g := −f and arguments as in the preceding example, we see that if x0 ∈ ∂f (x) we also have −x0 ∈ ∂g(x), hence x ∈ ∂g ∗ (−x0 ) = ∂(−f∗ ◦ (−IX ))(−x0 ) = ∂f∗ (x0 ).

10.6 The Fenchel—Rockafellar Duality A particular case requires some developments. It concerns the case when W and X are n.v.s. with dual spaces W 0 and X 0 , respectively, and when a subset K of W × X × X 0 × W 0 × R and a densely defined linear mapping A : X → W with closed graph and transpose A| are given such that J := {(x, x0 , r) : ∃u0 ∈ X 0 , w0 ∈ W 0 , (0W , x, u0 , w0 , r) ∈ K, x0 = u0 + A| w0 }. (10.2) Again, we consider W × X and X 0 × W 0 are paired with the coupling c of (10.1) which defines an isomorphism γ : (W × X)∗ → X 0 × W 0 . Thus the primal problem is (P)

find (x, r) ∈ X × R such that ∃w0 ∈ W 0 , (0W , x, −A| w0 , w0 , r) ∈ K.

The special case when K is the image by γ b := IW ×X × γ × IR of the subjet of a function k : W × X → R, A is continuous and j(x) := k(Ax, x),

∂j(x) = ∂k(Ax, x) ◦ (A, IX )

deserves some interest and illustrates what follows. More explicitly, in such a case one has K := {(w, x, x0 , w0 , r) : (x0 , w0 ) ∈ ∂k(w, x), r = k(w, x)}. This case is considered later on. Let us note that when K is the subjet of such a function k and when ∂ satisfies condition (C) the set J contains the subjet of j. But one may have J 6= J ∂ j when j = k ◦ (A, IX ). For j of this form, a natural perturbation of j is given by p(w, x) := k(w + Ax, x) for (w, x) ∈ W × X. Such a perturbation may inspire a hyperperturbation P in the general case to which we return. Given A, K, and J as in (10.2), we can introduce P by setting P := {(w, x, u0 + A| w0 , w0 , r) : u0 ∈ X 0 , w0 ∈ W 0 , (Ax + w, x, u0 , w0 , r) ∈ K} = {(w, x, x0 , w0 , r) : (Ax + w, x, x0 − A| w0 , w0 , r) ∈ K}.


365

Then J is the domain of the slice P0 : X × X 0 × R ⇒ W 0 of P given by P0 (x, x0 , r) := {w0 ∈ W 0 : (0W , x, x0 , w0 , r) ∈ P }, so that P is a hyperperturbation of J. The Ekeland transform P 0 := E(P ) ⊂ X 0 × W 0 × W × X × R of P is given by P 0 := {(u0 + A| w0 , w0 , w, x, hw0 , wi + hu0 + A| w0 , xi − r) : u0 ∈ X 0 , w0 ∈ W 0 , (Ax + w, x, u0 , w0 , r) ∈ K} 0 0 0 = {(x , w , w, x, r ) : (Ax + w, x, x0 − A| w0 , w0 , hw0 , wi + hx0 , xi − r0 ) ∈ K}, and the domain J 0 of the slice P00 : W 0 × W × R ⇒ X of P 0 defined by P00 (w0 , w, r0 ) := {x ∈ X : (0X 0 , w0 , w, x, r0 ) ∈ P 0 } is J 0 = {(w0 , w, r0 ) : ∃x ∈ X, (Ax + w, x, −A| w0 , w0 , hw0 , wi − r0 ) ∈ K} and the adjoint problem is (P 0 ) find (w0 , r0 ) ∈ W 0 × R such that ∃x ∈ X, (Ax, x, −A| w0 , w0 , −r0 ) ∈ K. Equivalently, because hw0 , Axi + h−A| w0 , xi = 0, we have (P 0 ) find (w0 , r0 ) ∈ W 0 ×R such that ∃x ∈ X, (−A| w0 , w0 , Ax, x, r0 ) ∈ E(K). Thus, (P 0 ) is obtained from E(K) in a way similar to the one (P) is deduced from K, with −A| , X 0 , W 0 , X, W substituted to A, W , X, W 0 , X 0 , respectively. When A is continuous, k is a generalized Legendre function, and K := γ b(J ∂ k) for some subdifferential ∂, one has (w0 , u0 ) ∈ ∂k(Ax + w, x) ⇔ (Ax + w, x) ∈ ∂kL (u0 , w0 )

so that P 0 is obtained from K 0 := γ b0 (J ∂ kL ) as P is obtained from K := J ∂ k 0 where γ b is a transposition similar to γ b. Then (P 0 ) is a substitute for the 0 0 extremization of the function j : w 7→ kL (−A| w0 , w0 ). Under appropriate assumptions, the preceding guideline becomes a precise result. Lemma 10.1. Given a function k : W × X → R∞ finite at (w, x) ∈ W × X and a continuous linear map A : X → W , let f : W × X → R∞ be given by f (w, x) := k(Ax + w, x). Then, for any subdifferential satisfying condition (D), one has (w, x, w0 , u0 + A| w0 , r) ∈ J ∂ f ⇔ (Ax + w, x, w0 , u0 , r) ∈ J ∂ k, so that P is the subjet of f up to a transposition.

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Proof. The result amounts to (w0 , u0 + A| w0 ) ∈ ∂f (w, x) ⇔ (w0 , u0 ) ∈ ∂k(Ax + w, x). It stems from condition (D), because the map B : (w, x) 7→ (Ax + w, x) is an isomorphism with inverse (z, x) 7→ (z − Ax, x), as a simple computation of the transpose B | of B shows. t u Proposition 10.4. Let W and X be reflexive Banach spaces with dual spaces W 0 and X 0 , respectively, and let A : X → W be linear and continuous. Let k : W × X → R∞ be a generalized Legendre function and let K := J ∂ k be its subjet. Then, for any subdifferential satisfying condition (D), the extremization problem (P 0 ) is the extremization problem associated with the hyperperturbation P 0 = J ∂ p0 of J 0 , where p0 (x0 , w0 ) = kL (w0 , x0 − A| w0 )

(x0 , w0 ) ∈ X 0 × W 0 .

Proof. Using the preceding lemma with a change of notation, we have (x, z − Ax) ∈ ∂p0 (x0 , w0 ) ⇔ (x, z) ∈ ∂kL (x0 − A| w0 , w0 ) ⇔ (x0 − A| w0 , w0 ) ∈ ∂k(x, z). Then, the definition of P 0 given above gives the result.

t u

Let us observe that when k is convex one gets the generalized Fenchel— Rockafellar duality (see for instance [47, Corollary 2.8.2]): inf k(Ax, x) = max (−k∗ (w0 , −A| w0 )) . 0 0

x∈X

w ∈W

Proposition 10.5. Let W and X be reflexive Banach spaces with dual spaces W 0 and X 0 , respectively, and let A : X → W be linear and continuous. Let k : W × X → R∞ be a l.s.c. proper convex function such that R+ (dom k∗ − (IW 0 , −A| )(W 0 ))

(10.3)

is a closed vector subspace of W 0 × X 0 . Then the extremization problem (P 0 ) coincides with the minimization problem minimize k∗ (w0 , −A| w0 )

w0 ∈ W 0 .

Proof. When k is a l.s.c. proper convex function, it is a generalized Legendre function and kL = k∗ , the Fenchel transform of k. Moreover, under the qualification condition (10.3), the Attouch—Brézis theorem ensures that for the convex function j 0 : w0 7→ kL (w0 , −A| w0 ) one has ∂j 0 (w0 ) = {w − Ax : (w, x) ∈ ∂k∗ (w0 , −A| w0 )} = {w − Ax : (w0 , −A| w0 ) ∈ ∂k(w, x)}.

t u


367

The next result deals with the particular case in which k(w, x) = g(w−b)+ f (x) for (w, x) ∈ W × X, where f : X → R∞ , g : W → R∞ are l.s.c. proper convex functions and b ∈ W is fixed. It follows from an easy computation of k∗ : k∗ (w0 , x0 ) = g ∗ (w0 ) + hw0 , bi + f ∗ (x0 ). Then one obtains that condition (10.3) is satisfied if and only if R+ (dom f ∗ + A| (W 0 )) is a closed vector subspace of X 0 . Corollary 10.1. Let W and X be reflexive Banach spaces with dual spaces W 0 and X 0 , respectively, let A : X → W be linear and continuous, and let f : X → R∞ , g : W → R∞ be l.s.c. proper convex functions such that R+ (dom f ∗ + A| (W 0 ))

(10.4)

is a closed vector subspace of W 0 . Then the extremization problem (P 0 ) coincides with the minimization problem minimize f ∗ (−A| w0 ) + g ∗ (w0 ) + hw0 , bi

w0 ∈ W 0 .

Let us note that when R+ (dom k − k ◦ (A, IX )(X)) is a closed vector subspace of W × X, the set J is the subjet of the function j, so that the situation is entirely symmetric. However, such a condition is not required to apply the duality relationships described in the preceding results.

10.7 The Toland Duality In [15] Ekeland applies his duality scheme to the case of the Toland duality. The primal problem is (T )

ext f (x) − g(Ax) x ∈ X,

where g : W → R and f : X → R∞ are l.s.c. proper convex functions and A : X → W is a continuous linear map. We interpret it as the extremization of the set J := {(x, x0 , r) ∈ X × X 0 × R : ∃w0 ∈ ∂g(Ax), ∃u0 ∈ ∂f (x), x0 = u0 − A| w0 }. However, we do not claim that J is the subjet of j : x 7→ f (x) − g(Ax). Thus, instead of using the subjet of k : (w, x) 7→ f (x) − g(w), we introduce the sets K := {(w, x, x0 , w0 , r) : −w0 ∈ ∂g(w), x0 ∈ ∂f (x), r = f (x) − g(w)}. Now we set P := {(w, x, x0 , w0 , r) : w0 ∈ ∂g(Ax − w), ∃u0 ∈ ∂f (x), x0 = u0 − A| w0 }

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which can be thought of as a similar interpretation of the subjet of p : (w, x) 7→ f (x) − g(Ax − w). Moreover, P := {(w, x, u0 + A| w0 , w0 , r) : u0 ∈ X 0 , w0 ∈ W 0 , (Ax − w, x, u0 , w0 , r) ∈ K} = {(w, x, x0 , w0 , r) : (Ax − w, x, x0 − A| w0 , w0 , r) ∈ K}. Then J is the domain of the slice P0 : X × X 0 × R ⇒ W 0 of P given by P0 (x, x0 , r) := {w0 ∈ W 0 : (0W , x, x0 , w0 , r) ∈ P }, so that P is a hyperperturbation of J. The Ekeland transformed set P 0 := E(P ) ⊂ X 0 × W 0 × W × X × R of P is given by P 0 := {(x0 , w0 , w, x, r0 ) : (Ax − w, x, x0 − A| w0 , w0 , hw0 , wi + hx0 , xi − r0 ) ∈ K} = {(u0 + A| w0 , w0 , w, x, hw0 , wi + hu0 + A| w0 , xi − r) : (Ax − w, x, u0 , w0 , r) ∈ K} and the domain J 0 of the slice P00 : W 0 × W × R ⇒ X of P 0 defined by P00 (w0 , w, r0 ) := {x ∈ X : (0X 0 , w0 , w, x, r0 ) ∈ P 0 } is J 0 = {(w0 , w, r0 ) : ∃x ∈ X, (Ax − w, x, −A| w0 , w0 , hw0 , wi − r0 ) ∈ K}. Thus the adjoint problem is (P 0 ) find (w0 , r0 ) ∈ W 0 × R such that ∃x ∈ X, (Ax, x, −A| w0 , w0 , −r0 ) ∈ K. We observe that, because hx0 − A| w0 , xi + hw0 , Ax − wi = hw0 , −wi + hx0 , xi, by the Fenchel equality hw0 , wi + hx0 , xi − f (x) + g(Ax + w) = hx0 − A| w0 , xi − h−w0 , Ax + wi − f (x) + g(Ax + w) = g ∗ (w0 ) − f ∗ (A| w0 − x0 ). Introducing the set K 0 := E(K), K 0 = {(x0 , w0 , w, x, r0 ) : (−w0 , x0 ) ∈ ∂g(w) × ∂f (x), r0 = h(w0 , x0 ), (w, x)i − f (x) + g(w)} = {(x0 , w0 , w, x, r0 ) : w ∈ ∂g ∗ (−w0 ), x ∈ ∂f ∗ (x0 ), r0 = f ∗ (x0 ) − g ∗ (−w0 )} which corresponds to the subjet of (x0 , w0 ) 7→ f ∗ (x0 ) − g ∗ (−w0 ), we have


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P 0 = {(x0 , w0 , w, x, r0 ) : (x0 − A| w0 , w0 , x, w, r0 ) ∈ K 0 } = {(x0 , w0 , w, x, f ∗ (x0 − A| w0 ) − g ∗ (−w0 )) : Ax − w ∈ ∂g ∗ (−w0 ), x ∈ ∂f ∗ (x0 − A| w0 )}, J 0 := {(w0 , w, r0 ) : ∃x ∈ X, Ax − w ∈ ∂g ∗ (−w0 ), x ∈ ∂f ∗ (−A| w0 ), r0 = f ∗ (−A| w0 ) − g ∗ (−w0 )} = {(w0 , w, r0 ) : ∃u ∈ ∂e g (A| w0 ), w + Au ∈ ∂ fe(w0 ), r0 = fe(w0 ) − ge(A| w0 )},

where fe(w0 ) := g ∗ (−w0 ), ge(x0 ) := f ∗ (−x0 ). Therefore, replacing fe, ge, A by g ∗ , f ∗ , A| , and using a construction similar to the one we have used to pass from j to J, the adjoint problem can be interpreted as (T 0 )

ext fe(w0 ) − ge(A| w0 ) w0 ∈ W 0 .

This is the Toland duality. Note that if we use a subdifferential ∂ such that ∂(−g)(x) = −∂g(x) for a convex function g, and if we dispose of regularity assumptions ensuring a sum rule, the preceding constructions are no more formal.

10.8 The Wolfe Duality Let us give a version of the Wolfe duality [46, 12, 34, 35, 36] that involves a family of minimization problems rather than a single one; we show that it can be interpreted as an instance of the Ekeland duality. Given a set U , n.v.s. W , X, a closed convex cone C in W , and mappings f : U × X → R and g : U × X → W which are differentiable in their second variable, let us consider the constrained optimization problem (M) minimize f (u, x) under the constraint g(u, x) ∈ C. We consider (M) as a minimization problem with respect to a primary variable x and a second variable u or as a family of partial minimization problems (Mu ) minimize fu (x) under the constraint gu (x) ∈ C,

u ∈ U.

The variant of the Wolfe dual we deal with is the family of partial maximization problems indexed by u ∈ U , (Wu ) maximize

u (x, y)

over (x, y) ∈ X ×Y subject to

∂ ∂x

u (x, y)

= 0, y ∈ C 0,

where u (x, y) := fu (x) + hy, gu (x)i is the classical Lagrangian, Y is the dual of W , and C 0 := {y ∈ Y : ∀w ∈ C hy, wi ≤ 0}. We observe that in (Wu ) the

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implicit constraint g(u, x) ∈ C which is difficult to deal with has disappeared, and an easier equality constraint appears. Then one has the following result, whose proof is similar to the one in [12, Theorem 4.7.1]. Theorem 10.2. Suppose that for all u ∈ U and all y ∈ −C 0 the functions fu and y ◦ gu are convex. Then, for all u ∈ U one has the weak duality relation sup(Wu ) ≤ inf(Mu ). If (M) has a solution, then there exists some u ∈ U such that strong duality holds; that is, the preceding inequality is an equality. In order to relate this result to the Ekeland scheme, for u ∈ U we introduce the subset Ju := {(x, y, x0 , y 0 , r) : r = fu (x), gu (x) ∈ C, x0 = fu0 (x) + y ◦ gu0 (x), y 0 = gu (x), hy, gu (x)i = 0} of X × C 0 × X 0 × W × R, so that Ju is the intersection of {(x, y, x0 , y 0 , r) ∈ X × C 0 × X 0 × W × R : hy, gu (x)i = 0} with the one-jet J1

u

:= {(x, y, x0 , y 0 , r) : (x0 , y 0 ) = D u (x, y), r =

u (x, y)}

of the function u . The extremization of Ju consists in searching pairs (x, y) ∈ X × C 0 which are critical points of u with respect to X × C 0 , that is, which satisfy ∂ ∂x

u (x, y)

= 0,

∂ ∂y

u (x, y)

∈ C 00 = C,

hy,

∂ ∂y

u (x, y)i

= 0.

This is exactly the set of solutions of the Kuhn—Tucker system. It is natural to associate with (Mu ) the perturbed problem by w ∈ W (Mu,w ) minimize fu (x) under the constraint gu (x) + w ∈ C. We associate with this problem the subset P of the set W × gu−1 (C) × C 0 × X 0 × Y 0 × W 0 × R given by (w, x, y, x0 , y 0 , w0 , r) ∈ P ⇔ x0 = Dfu (x)+y ◦ Dgu (x), y 0 = gu (x)+w, w0 = y, r = fu (x)+hy, gu (x)+wi. It is clearly a hyperperturbation of Ju . A short computation shows that its Ekeland transform P 0 is characterized by (w0 , x0 , y 0 , w, x, y, r0 ) ∈ P 0 if and only if (w, x, y, x0 , y 0 , w0 , r0 ) ∈ W × g −1 (C) × C 0 × X 0 × Y 0 × W 0 × R and r0 = hw0 , wi+hx0, xi−fu (x), w0 = y, x0 = Dfu (x)+y◦Dgu (x), y 0 = gu (x)+w.


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Thus, considering w as a parameter and (x, y) as the decision variable, we can set Ju0 = {(w0 , w, r0 ) : ∃(x, y) ∈ gu−1 (C) × C 0 , (w0 , 0X 0 , 0Y 0 , w, x, y, r0 ) ∈ P 0 }. We obtain that (w0 , w, r0 ) ∈ Ju0 if and only if there exists (x, y) ∈ gu−1 (C)×C 0 such that y := w0 , Dfu (x) + y ◦ Dgu (x) = 0, hy, gu (x) + wi = 0,

gu (x) + w ∈ C, r0 = hw0 , wi − fu (x).

Then r0 = hy, −gu (x)i − fu (x) = − u (x, y). We see that ext(M0u ) corresponds to the search of (w0 , r0 , x) ∈ Y × R × X such that Dfu (x) + y ◦ Dgu (x) = 0, hw0 , gu (x)i = 0,

gu (x) ∈ C, r0 = − u (x, y),

y ∈ C0,

or, in other terms, to the search of (x, y, r0 ) ∈ gu−1 (C) × C 0 × R such that ∂ u (x, y)/∂x = 0, ∂ u (x, y)/∂y ∈ C, r0 = − u (x, y): (y, r0 ) ∈ ext(M0u ) ⇔ ∃x ∈ X : gu (x) ∈ C, hy, gu (x)i = 0, Dfu (x) + y ◦ Dgu (x) = 0,

r0 = − u (x, y).

Now (x, y) is a critical point for the problem (M0u ) maximize

u (x, y)

over (x, y) ∈ X × Y ∂ under the constraints gu (x) ∈ C, u (x, y) = 0 ∂x

if and only if there exist multipliers y ∈ C 0 , x∗∗ ∈ X ∗∗ such that for all (b x, yb) ∈ X × Y , hy, gu (x)i = 0,

x, yb) + hy, Dgu (x)(b x)i + hx∗∗ , D −D u (x, y)(b

∂ ∂x

x, yb)i u (x, y)(b

= 0.

Taking y = 0, x∗∗ = 0, we see that for any solution (y, r0 ) of ext(M0u ) and for any x ∈ X satisfying the requirements of ext(M0u ), one gets a critical point (x, y) of the problem (M0u ). In turn, considering (u, x) as an auxiliary variable and y as the primary variable, one is led to the maximization problem (Wu ). However, a solution (x, y) of (Wu ) should satisfy the extra conditions gu (x) ∈ C, hy, gu (x)i = 0 in order to yield a solution to ext(M0u ). Note that in the case of the quadratic problem (Q)

minimize

1 hQx, xi + hq, xi 2

subject to Ax − b ∈ C,

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where Q : X → X 0 is linear, continuous, and symmetric (but not necessarily semidefinite positive), A : X → W , q ∈ X 0 , b ∈ W , C being a closed convex cone of W , the Wolfe dual 1 hQx, xi + hq, xi + hy, Ax − bi over (x, y) ∈ X × Y 2 subject to Qx + q + y ◦ A = 0

(W) maximize

is a simple quadratic problem with linear constraints. It can be given necessary and sufficient optimality conditions provided the map (x, y) 7→ Qx+y◦A has a closed range in X 0 .

10.9 The Clarke Duality Let X be a reflexive Banach space, let A : X → X ∗ be a densely defined selfadjoint operator (i.e., such that hAx1 , x2 i = hx1 , Ax2 i for any x1 , x2 ∈ dom A) and let g : X → R ∪ {+∞} be a l.s.c. proper convex function. Let X 0 := X ∗ and let J be given by J := {(x, x0 , r) ∈ X × X 0 × R : x0 + Ax ∈ ∂g(x), r = j(x)} where 1 j(x) := g(x) − hAx, xi for x ∈ dom A ∩ dom g, j(x) = +∞ else. 2 Let us consider the extremization problem of J: (P)

find (x, r) ∈ X × R such that Ax ∈ ∂g(x), r = j(x).

Here we have taken −A instead of A as in [7, 16] and elsewhere in order to get a more symmetric form of the result; of course, this choice is inessential as we make no positiveness assumption on A. When A is continuous, and when the subdifferential ∂ satisfies condition (T) (in particular for the Fréchet, the Hadamard, the moderate, and the Clarke subdifferentials) J is the subjet of j because in that case one has x0 ∈ ∂j(x) ⇔ x0 + Ax ∈ ∂g(x). In particular, x is a critical point of j in the sense 0 ∈ ∂j(x) iff Ax ∈ ∂g(x). Then (P) corresponds to the extremization of j. Let us introduce a hyperperturbation of J by setting W := X ∗ , W 0 := X, 0 X := X ∗ , and P := {(w, x, x0 , x, j(x)) ∈ W × dom j × X 0 × W 0 × R : x0 + Ax − w ∈ ∂g(x)}.


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In fact, we have P0 (x, x0 , r) := {w0 ∈ W 0 : (0W , x, x0 , w0 , r) ∈ P } = {w0 ∈ W 0 : w0 = x, x0 + Ax ∈ ∂g(x), r = j(x)}, hence (x, x0 , r) ∈ dom P0 ⇔ x0 + Ax ∈ ∂g(x), r = j(x) ⇔ (x, x0 , r) ∈ J, so that P is indeed a hyperperturbation of J in the sense given above. Although we do not need the following result to proceed, it may serve as a guide line. Lemma 10.2. When A is continuous and ∂ satisfies conditions (F), (P), (T), the set P is the subjet of the function f : W × X → R∞ given by 1 f (w, x) = g(x) − hAx, xi + hw, xi 2 and f is an Ekeland function. Proof. When A is continuous f is the sum of the continuously differentiable function (w, x) 7→ − 12 hAx, xi+hw, xi with the convex function (w, x) 7→ g(x), and conditions (T), (P), and (F) ensure that (w0 , x0 ) ∈ ∂f (w, x) ⇔ w0 = x, x0 + Ax − w ∈ ∂g(x).

(10.5)

−1

Then, for (w0 , x0 ) ∈ W 0 × X 0 and for (w, x) ∈ (∂f ) (w0 , x0 ) one has µ ¶ 1 f E (w0 , x0 ) = hw, w0 i + hx, x0 i − g(x) − hAx, xi + hw, xi 2 1 = hw0 , x0 i + hAw0 , w0 i − g(w0 ) 2 and we see that this value does not depend on the choice of (w, x) ∈ −1 t u (∂f ) (w0 , x0 ): f is an Ekeland function. Let us return to the general case. In order to describe the dual problem (P 0 ), we observe that J 0 = {(w0 , w, r0 ) ∈ W 0 × W × R : ∃x ∈ X, (0X 0 , w0 , w, x, r0 ) ∈ P 0 } = {(w0 , w, r0 ) ∈ W 0 × W × R : ∃x ∈ X, (w, x, 0X 0 , w0 , hw, w0 i − r0 ) ∈ P } and x ∈ P00 (w0 , 0W , r0 ) ⇔ (0W , x, 0X 0 , w0 , −r0 ) ∈ P

so that (w0 , 0W , r0 ) ∈ J 0 = dom P00 iff there exists some x ∈ dom j ⊂ X such that Ax ∈ ∂g(x), x = w0 , r0 = −f (0, x). Thus, because g is convex and A is symmetric,

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(w0 , r0 ) ∈ ext J 0 ⇔ w0 ∈ dom j, Aw0 ∈ ∂g(w0 ), r0 = −f (0, w0 ) ⇔ w0 ∈ dom j, w0 ∈ ∂g ∗ (Aw0 ), r0 = −j(w0 ) ⇒ w0 ∈ dom j, Aw0 ∈ A (∂g ∗ (Aw0 )) ⊂ ∂ (g ∗ ◦ A) (w0 ), r0 = −j(w0 ). In particular, when ∂ satisfies conditions (F) and (T) and A is continuous, for any (w0 , r0 ) ∈ ext J 0 , the pair (w0 , −r0 ) is a critical pair of the function j 0 : X → R ∪ {+∞} given by 1 j 0 (x) := g ∗ (Ax) − hAx, xi. 2 This function is invariant by addition of an element of Ker A, thus we have obtained under these conditions the first part of the following statement which subsumes Clarke duality. In order to prove the second part we introduce the function j 00 given by 1 ∗ j 00 (x) := (g ∗ ◦ A) (Ax) − hAx, xi. 2 Theorem 10.3. Suppose g is l.s.c. proper convex, ∂ satisfies (F), (T), and A is continuous. Then, (a) For any critical pair (x, r) of J and for any u ∈ Ker A, the pair (x + u, −r) is a critical pair of J 0 . (b) For any critical pair (x0 , r0 ) of J 0 and for any u ∈ Ker A, the pair 0 (x + u, −r0 ) is a critical pair of j 00 . If moreover g is convex and R+ (dom g ∗ − A(X)) = X 0 , then there exists u0 ∈ Ker A such that (x0 + u, −r0 ) is a critical pair of j.

Proof. Because J 0 has the same form as J, with g replaced by g ∗ ◦A, we obtain from part (a) that for any critical pair (x0 , r0 ) of j 0 and for any u ∈ Ker A, the pair (x0 + u, −r0 ) is a critical pair of 1 x 7→ (g ∗ ◦ A)∗ (Ax) − hAx, xi = j 00 (x). 2 On the other hand, x0 is a critical point of j 0 means that Ax0 ∈ ∂(g ∗ ◦ A)(x0 ). Now, under condition (C), the Attouch—Brézis theorem ensures the equalities ∂ (g ∗ ◦ A) (x0 ) = A| (∂g ∗ (Ax0 )) = A(∂g ∗ (Ax0 )), so that there exists some y 0 ∈ ∂g ∗ (Ax0 ) such that Ax0 = Ay 0 . Thus, one has u0 := y 0 − x0 ∈ KerA and because y 0 ∈ ∂g ∗ (Ax0 ), by the reciprocity formula, we get Ax0 ∈ ∂g(y 0 ) or Ay 0 ∈ ∂g(y 0 ). Therefore, (x0 + u, −r0 ) is a critical pair of j. t u


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References 1. Amahroq, T., Penot, J.-P., and Syam, A., Subdifferentiation and minimization of the difference of two functions, Set-Valued Anal. (to appear) DOI: 10.1007/s11228-0080085-9. 2. Aubin, J.-P., and Ekeland, I., Applied Nonlinear Analysis, Wiley, New York (1984). 3. Aubin, J.-P., and Frankowska, H., Set-Valued Analysis, Birkh¨ auser, Boston (1990). 4. Aussel, D., Corvellec, J.-N., and Lassonde, M., Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc. 347, No. 10, 4147—4161 (1995). 5. Blot, J., and Az´ e, D., Syst` emes Hamiltoniens: Leurs Solutions P´ eriodiques, Textes Math´ ematiques Recherche 5, Cedic/Nathan, Paris (1982). 6. Clarke, F., Optimization and Nonsmooth Analysis, Wiley (1983), SIAM, Philadelphia (1990). 7. Clarke, F., A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc. 76, 186—188 (1979). 8. Clarke, F.H., Periodic solutions to Hamiltonian inclusions, J. Diff. Equations 40, 1—6 (1981). 9. Clarke, F.H., On Hamiltonian flows and symplectic transformations, SIAM J. Control Optim. 20, 355—359 (1982). 10. Clarke, F., and Ekeland, I., Hamiltonian trajectories having prescribed minimal period, Commun. Pure Appl. Math. 33, 103—116 (1980). 11. Collier, J.B., The dual of a space with the Radon-Nikodym property, Pacific J. Math. 64, 103—106 (1976). 12. Craven, B.D., Mathematical Programming and Control Theory, Chapman & Hall, London (1978). 13. Dorn, W.S., Duality in quadratic programming, Quart. Appl. Math. 18, 155—162 (1960). 14. Ekeland, I., Legendre duality in nonconvex optimization and calculus of variations, SIAM J. Control Optim. 15, No. 6, 905—934 (1977). 15. Ekeland, I., Nonconvex duality, Bull. Soc. Math. France M´ emoire No. 60, Analyse Non Convexe, Pau, 1977, 45—55 (1979). 16. Ekeland, I., Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Math. 19, Springer-Verlag, Berlin (1990). 17. Ekeland, I., and Hofer, H., Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Invent. Math. 81, 155—188 (1985). 18. Ekeland, I., and Lasry, J.-M., Principes variationnels en dualité, C.R. Acad. Sci. Paris 291, 493—497 (1980). 19. Ekeland, I., and Lasry, J.-M., On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (2) 112, 283—319 (1980). 20. Ekeland, I., and Lasry, J.-M., Duality in nonconvex variational problems, in Advances in Hamiltonian Systems, Aubin, Bensoussan, and Ekeland, eds., Birkh¨ auser, Basel (1983). 21. Frenk, J.B.G., and Schaible, S., Fractional programming, in Handbook of Generalized Convexity and Generalized Monotonicity, Hadjisavvas, N., Koml´ osi, S., and Schaible, S., eds., Nonconvex Optimization and Its Applications 76, Springer, New York, 335— 386 (2005). 22. Gao, D.Y., Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optim. 17, No. 1—4, 127—160 (2000). 23. Gao, D.Y., Duality Principles in Nonconvex Systems: Theory, Methods and Applications, Nonconvex Optimization and Its Applications 39, Kluwer, Dordrecht (2000). 24. Gao, D.Y., Complementarity, polarity and triality in non-smooth, non-convex and non-conservative Hamilton systems, Phil. Trans. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 359, No. 1789, 2347—2367 (2001).

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25. Gao, D.Y., Perfect duality theory and complete solutions to a class of global optimization problems, Optimization 52, No. 4—5, 467—493 (2003). 26. Gao, D.Y., Complementary, Duality and Symmetry in Nonlinear Mechanics: Proceedings of the IUTAM Symposium, Shanghai, China, August 13—16, 2002, Advances in Mechanics and Mathematics 6, Kluwer, Boston, MA (2004). 27. Gao, D.Y., Canonical duality theory and solutions to constrained nonconvex quadratic programming, J. Global Optim. 29, No. 3, 377—399 (2004). 28. Gao, D.Y., Ogden, R.W., and Stavroulakis, G., eds., Nonsmooth/Nonconvex Mechanics: Modeling, Analysis and Numerical Methods, Nonconvex Optimization and Its Applications 50, Kluwer, Boston (2001). 29. Gao, D.Y., and Teo, K.L., eds., Special issue: On duality theory, methods and applications, J. Global Optim. 29, No. 4, 335—516 (2004). 30. Ioffe, A.D., On the local surjection property, Nonlinear Anal. Theory Meth. Appl. 11, 565—592 (1987). 31. Ioffe, A.D., Approximate subdifferentials and applications, III: The metric theory, Mathematika 36, No. 1, 1—38 (1989). 32. Ioffe, A.D., Metric regularity and subdifferential calculus, Russ. Math. Surv. 55, No. 3, 501—558 (2000); translation from Usp. Mat. Nauk 55, No. 3, 103—162 (2000). 33. Michel, P., and Penot, J.-P., A generalized derivative for calm and stable functions, Differential Integral Equat. 5, No. 2, 433—454 (1992). 34. Mititelu, S., The Wolfe duality without convexity, Stud. Cercet. Mat. 38, 302—307 (1986). 35. Mititelu, S., Hanson’s duality theorem in nonsmooth programming, Optimization 28, No. 3—4, 275—281 (1994). 36. Mittelu, S., Conditions de Kuhn-Tucker et dualit´ e de Wolfe dans la programmation non lipschitzienne, Bull. Math. Soc. Sci. Math. Roum. Nouv. S´ er. 37, No. 1—2, 65—74 (1993). 37. Penot, J.-P., Favorable classes of mappings and multimappings in nonlinear analysis and optimization, J. Convex Anal. 3, No. 1, 97—116 (1996). 38. Penot, J.-P., Subdifferential calculus without qualification assumptions, J. Convex Anal. 3, No. 2, 1—13 (1996). 39. Penot, J.-P., Mean-value theorem with small subdifferentials, J. Optim. Theory Appl. 94, No. 1, 209—221 (1997). 40. Penot, J.-P., Mean value theorems for mappings and correspondences, Acta Math. Vietnamica 26, No. 3, 365—376 (2002). 41. Penot, J.-P., Unilateral analysis and duality, in Essays and Surveys in Global Optimization, Savard, G., et al., eds., Springer, New York, 1—37 (2005). 42. Penot, J.-P., Legendre functions and the theory of characteristics, preprint, University of Pau (2004). 43. Penot, J.-P., The Legendre transform of correspondences, Pacific J. Optim. 1, No. 1, 161—177 (2005). 44. Penot, J.-P., Critical duality, J. Global Optim. 40, No. 1—3, 319—338 (2008). 45. Penot, J.-P., and Rubinov, A., Multipliers and general Lagrangians, Optimization 54, No. 4—5, 443—467 (2005). 46. Wolfe, P., A duality theorem for non-linear programming, Quart. Appl. Math. 19, 239—244 (1961). 47. C. Z˘ alinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).

Chapter 11

Global Optimization in Practice: State of the Art and Perspectives János D. Pintér

Summary. Global optimization–the theory and methods of finding the best possible solution in multiextremal models–has become a subject of interest in recent decades. Key theoretical results and basic algorithmic approaches have been followed by software implementations that are now used to handle a growing range of applications. This work discusses some practical aspects of global optimization. Within this framework, we highlight viable solution approaches, modeling environments, software implementations, numerical examples, and real-world applications. Key words: Nonlinear systems analysis and management, global optimization strategies, modeling environments and global solver implementations, numerical examples, current applications and future perspectives

11.1 Introduction Nonlinearity plays a fundamental role in the development of natural and man-made objects, formations, and processes. Consequently, nonlinear descriptive models are of key relevance across the range of quantitative scientific studies. For related discussions that illustrate this point consult, for instance, Bracken and McCormick (1968), Rich (1973), Mandelbrot (1983), Murray (1983), Casti (1990), Hansen and Jørgensen (1991), Schroeder (1991), Bazaraa et al. (1993), Stewart (1995), Grossmann (1996), Pardalos et al. (1996), Pintér (1996a, 2006a, 2009), Aris (1999), Bertsekas (1999), Corliss and Kearfott (1999), Floudas et al. (1999), Gershenfeld (1999), Papalambros and Wilde (2000), Chong and Zak (2001), Edgar et al. (2001), Gao et J´ anos D. Pintér Pint´ er Consulting Services, Inc., Canada, and Bilkent University, Turkey E-mail: [email protected]; Web site: www.pinterconsulting.com D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_11, © Springer Science+Business Media, LLC 2009

377

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J´ anos D. Pint´ er

al. (2001), Jacob (2001), Pardalos and Resende (2002), Schittkowski (2002), Tawarmalani and Sahinidis (2002), Wolfram (2002), Diwekar (2003), Stojanovic (2003), Zabinsky (2003), Bornemann et al. (2004), Fritzson (2004), Neumaier (2004), Bartholomew-Biggs (2005), Hillier and Lieberman (2005), Lopez (2005), Nowak (2005), Kampas and Pintér (2009), as well as many other topical works. Decision support (control, management, or optimization) models that incorporate an underlying nonlinear system description frequently have multiple–local and global–optima. The objective of global optimization (GO) is to find the “absolutely best” solution of nonlinear optimization models under such circumstances. We consider the general continuous global optimization (CGO) model defined by the following ingredients. • • • •

x l, u f (x) g(x)

decision vector, an element of the real Euclidean n-space Rn explicit, finite n-vector bounds of x that define a “box” in Rn continuous objective function, f : Rn → R m-vector of continuous constraint functions, g : Rn → Rm

Applying this notation, the CGO model is stated as min f (x)

(11.1)

x ∈ D := {x : l ≤ x ≤ u g(x) ≤ 0}.

(11.2)

In (11.2) all vector inequalities are interpreted componentwise (l, x, u, are n-component vectors and the zero denotes an m-component vector). The set of the additional constraints g could be empty, thereby leading to boxconstrained GO models. Let us also note that formally more general optimization models that also include = and ≥ constraint relations and/or explicit lower bounds on the constraint function values can be simply reduced to the model form (11.1) and (11.2). The CGO model is very general: in fact, it evidently subsumes linear programming and convex nonlinear programming models, under corresponding additional specifications. Furthermore, CGO also subsumes (formally) the entire class of pure and mixed integer programming problems. To see this, notice that all bounded integer variables can be represented by a corresponding set of binary variables, and then every binary variable y ∈ {0, 1} can be equivalently represented by its continuous extension y ∈ [0, 1] and the nonconvex constraint y(1 − y) ≤ 0. Of course, this reformulation approach may not be best–or even suitable–for “all” mixed integer optimization models: however, it certainly shows the generality of the CGO model framework. Without going into details, note finally that models with multiple (partially conflicting) objectives are also often deduced to suitably parameterized collections of CGO (or simpler optimization) models: this remark also hints at the interchangeability of the objective f and one of the (active) constraints from g.

11 Global Optimization in Practice

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Let us observe next that if D is nonempty, then the above-stated basic analytical assumptions guarantee that the optimal solution set X ∗ in the CGO model is nonempty. This result directly follows by the classical theorem of Weierstrass that states the existence of the global minimizer point–or, in general, a set of such points–of a continuous function over a nonempty, bounded, and closed (compact) set. For reasons of numerical tractability, the following additional requirements are also often postulated. • D is a full-dimensional subset (“body”) in Rn . • The set of globally optimal solutions to (11.1) and (11.2) is at most countable. • f and g (componentwise) are Lipschitz-continuous functions on [l, u]. Without going into technical details, notice that the first of these assumptions (the set D is the closure of its nonempty interior) makes algorithmic search easier (or at all possible) within D. The second assumption supports theoretical convergence results: note that in most well-posed practical GO problems the set of global optimizers consists either of a single point x∗ or at most of several points. The third assumption is a sufficient condition for estimating f ∗ = f (x∗ ) on the basis of a finite set of generated feasible search points. (Recall that the real-valued function h is Lipschitz-continuous on its domain of definition D ⊂ Rn , if |h(x1 ) − h(x2 )| ≤ Lkx1 − x2 k holds for all pairs x1 ∈ D, x2 ∈ D; here L = L(D, h) is a suitable Lipschitz-constant of h on the set D.) We emphasize that the exact knowledge of the smallest suitable Lipschitz-constant for each model function is not required, and in practice such information is typically unavailable. At the same time, all models defined by continuously differentiable functions f and g belong to the CGO or even to the Lipschitz model-class. The notes presented above imply that the CGO model-class covers a very broad range of optimization problems. As a consequence of this generality, it includes also many model instances that are difficult to solve numerically. For illustration, a merely one-dimensional, box-constrained GO model based on the formulation (11.3) is shown in Figure 11.1. min cos(x) sin(x2 − x)

0 ≤ x ≤ 10.

(11.3)

Model complexity often increases dramatically (in fact, it can grow exponentially) as the model size expressed by the number of variables and constraints grows. To illustrate this point, Figure 11.2 shows the objective function in the model (11.4) that is simply generalized from (11.3) as min cos(x) sin(y 2 −x)+cos(y) sin(x2 −y)

0 ≤ x ≤ 10, 0 ≤ y ≤ 10. (11.4)

The presented two (low-dimensional, and only box-constrained) models already indicate that GO models–for instance, further extensions of model (11.3), perhaps with added complicated nonlinear constraints–could become

380

Fig. 11.1 The objective function in model (11.3).

Fig. 11.2 The objective function in model (11.4).



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truly difficult to handle numerically. One should also point out here that a direct analytical solution approach is viable only in very special cases, because in general (under further structural assumptions) one should investigate all Kuhn—Tucker points (minimizers, maximizers, and saddle points) of the CGO model. (Think of carrying out this analysis for the model depicted in Figure 11.2, or for its 100-dimensional extension.) Arguably, not all GO models are as difficult as indicated by Figures 11.1 and 11.2. At the same time, we typically do not have the possibility to directly inspect, visualize, or estimate the overall numerical difficulty of a complicated nonlinear (global) optimization model. A practically important case is when one needs to optimize the parameters of a model that has been developed by someone else. The model may be confidential, or just visibly complex; it could even be presented to the optimization engine as a compiled (object, library, or similar) software module. In such situations, direct model inspection and structure verification are not possible. In other practically relevant cases, the evaluation of the optimization model functions may require the numerical solution of a system of embedded differential and/or algebraic equations, the evaluation of special functions, integrals, the execution of other deterministic computational procedures or stochastic simulation modules, and so on. Traditional nonlinear optimization methods (discussed in most topical textbooks such as Bazaraa et al., 1993, Bertsekas, 1999, Chong and Zak, 2001, and Hillier and Lieberman, 2005) search only for local optima. This generally followed approach is based on the tacit assumption that a “sufficiently good” initial solution (that is located in the region of attraction of the “true” global solution) is available. Figures 11.1 and 11.2 and the practical situations mentioned above suggest that this may not always be a realistic assumption. Nonlinear models with less “dramatic” difficulty, but in (perhaps much) higher dimensions may also require global optimization. For instance, in advanced engineering design, optimization models with hundreds, thousands, or more variables and constraints are analyzed and need to be solved. In similar cases, even an approximately completed, but genuinely global scope search strategy may (and typically will) yield better results than the most sophisticated local search approach “started from the wrong valley”. This fact has motivated research to develop practical GO strategies.

11.2 Global Optimization Strategies As of today, well over a hundred textbooks and an increasing number of Web sites are devoted (partly or completely) to global optimization. Added to this massive amount of information is a very substantial body of literature on combinatorial optimization (CO), the latter being, at least in theory, a “subset of GO.” The most important global optimization model types and (mostly exact, but also several prominent heuristic) solution approaches are

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discussed in detail by the Handbook of Global Optimization volumes, edited by Horst and Pardalos (1995), and by Pardalos and Romeijn (2002). We also refer to the topical Web site of Neumaier (2006), with numerous links to other useful information sources. The concise review of GO strategies presented here draws on these sources, as well as on the more detailed expositions in Pintér (2001a, 2002b). Let us point out that some of the methods listed below are more often used in solving CGO models, whereas others have been mostly applied so far to handle CO models. Because CGO formally includes CO, it should not be surprising that approaches suitable for certain specific CO model-classes can (or could) be put to good use to solve CGO models. Instead of a more detailed (but still not unambiguous) classification, here we simply classify GO methods into two primary categories: exact and heuristic. Exact methods possess theoretically established (deterministic or stochastic) global convergence properties. That is, if such a method could be carried out completely as an infinite iterative process, then the generated limit point(s) would belong to the set of global solutions X ∗ . (For a single global solution x∗ , this would be the only limit point.) In the case of stochastic GO methods, the above statement is valid “only” with probability one. In practice–after a finite number of algorithmic search steps–one can only expect a numerically validated or estimated (deterministic or stochastic) lower bound for the global optimum value z ∗ = f (x∗ ), as well as a best feasible or near-feasible global solution estimate. We emphasize that to produce such estimates is not a trivial task, even for implementations of theoretically wellestablished algorithms. As a cautionary note, one can conjecture that there is no GO method, and never will be one, that can solve “all” CGO models with a certain number of variables to an arbitrarily given precision (in terms of the argument x∗ ), within a given time frame, or within a preset model function evaluation count. To support this statement, please recall Figures 11.1 and 11.2: both of the objective functions displayed could be made arbitrarily more difficult, simply by changing the frequencies and amplitudes of the embedded trigonometric terms. We do not attempt to display such “monster” functions, because even the best visualization software will soon become inadequate: think for instance of a function such as 1000 cos(1000x) sin(1000(x2 − x)). For a more practically motivated example, one can also think of solving a difficult system of nonlinear equations: here, after a prefixed finite number of model function evaluations, we may not have an “acceptable” approximate numerical solution. Heuristic methods do not possess similar convergence guarantees to those of exact methods. At the same time, they may provide good quality solutions in many difficult GO problems, assuming that the method in question suits the specific model type (structure) solved. Here a different cautionary note is in order. Because such methods are often based on some generic metaheuristics, overly optimistic claims regarding the “universal” efficiency of their implementations are often not supported by results in solving truly difficult, especially nonlinearly constrained, GO models. In addition, heuris-


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tic metastrategies are often more difficult to adjust to new model types than some of the solver implementations based on exact algorithms. Exact stochastic methods based on direct sampling are a good example for the latter category, because these can be applied to “all” GO models directly, without a need for essential code adjustments and tuning. This is in contrast, for example, to most population-based search methods in which the actual steps of generating new trial solutions may depend significantly on the structure of the model-instance solved.

11.2.1 Exact Methods • “Na¨ıve” approaches (grid search, pure random search): these are obviously convergent, but in general “hopeless” as the problem size grows. • Branch-and-bound methods: these include interval-arithmetic-based strategies, as well as customized approaches for Lipschitz global optimization and for certain classes of difference of convex functions (D.C.) models. Such methods can also be applied to constraint satisfaction problems and to (general) pure and mixed integer programming. • Homotopy (path following, deformation, continuation, trajectory, and related other) methods: these are aimed at finding the set of global solutions in smooth GO models. • Implicit enumeration techniques: examples are vertex enumeration in concave minimization models, and generic dynamic programming in the context of combinatorial optimization. • Stochastically convergent sequential sampling methods: these include adaptive random searches, single- and multistart methods, Bayesian search strategies, and their combinations. For detailed expositions related to deterministic GO techniques in addition to the Handbooks mentioned earlier, consult, for example, Horst and Tuy (1996), Kearfott (1996), Pintér (1996a), Tawarmalani and Sahinidis (2002), Neumaier (2004), and Nowak (2005). On stochastic GO strategies, consult, for example, Zhigljavsky (1991), Boender and Romeijn (1995), Pintér (1996a), and Zabinsky (2003).

11.2.2 Heuristic Methods • Ant colony optimization is based on individual search steps and “ant-like” interaction (communication) between search agents. • Basin-hopping strategies are based on a sequence of perturbed local searches, in an effort to find improving optima.

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• Convex underestimation attempts are based on a limited sampling effort that is used to estimate a postulated (approximate) convex objective function model. • Evolutionary search methods model the behavioral linkage among the adaptively changing set of candidate solutions (“parents” and their “children,” in a sequence of “generations”). • Genetic algorithms emulate specific genetic operations (selection, crossover, mutation) as these are observed in nature, similarly to evolutionary methods. • Greedy adaptive search strategies (a metaheuristics often used in combinatorial optimization) construct “quick and promising” initial solutions which are then refined by a suitable local optimization procedure. • Memetic algorithms are inspired by analogies to cultural (as opposed to natural) evolution. • Neural networks are based on a model of the parallel architecture of the brain. • Response surface methods (directed sampling techniques) are often used in handling expensive “black box” optimization models by postulating and then gradually adapting a surrogate function model. • Scatter search is similar in its algorithmic structure to ant colony, genetic, and evolutionary searches, but without their “biological inspiration.” • Simulated annealing methods are based on the analogy of cooling crystal structures that will attain a (low-energy level, stable) physical equilibrium state. • Tabu search forbids or penalizes search moves which take the solution in the next few iterations to points in the solution space that have been previously visited. (Tabu search as outlined here has been typically applied in the context of combinatorial optimization.) • Tunneling strategies, filled function methods, and other similar methods attempt to sequentially find an improving sequence of local optima, by gradually modifying the objective function to escape from the solutions found. In addition to the earlier mentioned topical GO books, we refer here to several works that discuss mostly combinatorial (but also some continuous) global optimization models and heuristic strategies. For detailed discussions of theory and applications, consult, for example, Michalewicz (1996), Osman and Kelly (1996), Glover and Laguna (1997), Voss et al. (1999), Jacob (2001), Ferreira (2002), Rothlauf (2002), and Jones and Pevzner (2004). It is worth pointing out that Rudolph (1997) discusses the typically missing theoretical foundations for evolutionary algorithms, including stochastic convergence studies. (The underlying key convergence results for adaptive stochastic search methods are discussed also in Pintér (1996a).) The topical chapters in Pardalos and Resende (2002) also offer expositions related to both exact and heuristic GO approaches.


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To conclude this very concise review, let us emphasize again that numerical GO can be tremendously difficult. Therefore it can be good practice to try several–perhaps even radically different–search approaches to tackle GO models, whenever this is possible. To do this, one needs ready-to-use model development and optimization software tools.

11.3 Nonlinear Optimization in Modeling Environments Advances in modeling techniques, solver engine implementations and computer technology have led to a rapidly growing interest in modeling environments. For detailed discussions consult, for example, the topical Annals of Operations Research volumes edited by Maros and Mitra (1995), Maros et al. (1997), Vladimirou et al. (2000), Coullard et al. (2001), as well as the volumes edited by Voss and Woodruff (2002) and by Kallrath (2004). Additional useful information is provided by the Web sites of Fourer (2006), Mittelmann (2006), and Neumaier (2006), with numerous further links. Prominent examples of widely used modeling systems that are focused on optimization include AIMMS (Paragon Decision Technology , 2006), AMPL (Fourer et al., 1993), the Excel Premium Solver Platform (Frontline Systems , 2006), GAMS (Brooke et al., 1988), ILOG (2004), the LINDO Solver Suite (LINDO Systems, 2006), MPL (Maximal Software, 2006), and TOMLAB (2006). (Please note that the literature references cited may not always reflect the current status of the modeling systems discussed here: for the latest information, contact the developers and/or visit their Web sites.) There also exist a large variety of core compiler platform-based solver systems with more or less built-in model development functionality: in principle, such solvers can be linked to the modeling languages listed above. At the other end of the spectrum, there is also notable development in relation to integrated scientific and technical computing (ISTC) systems such as Maple (Maplesoft, 2006), Mathematica (Wolfram Research, 2006), Mathcad (Mathsoft, 2006), and MATLAB (The MathWorks, 2006). From among the many hundreds of books discussing ISTC systems, we mention here as examples the works of Birkeland (1997), Bhatti (2000), Parlar (2000), Wright (2002), Wilson et al. (2003), Moler (2004), Wolfram (2003), Trott (2004), and Lopez (2005). The ISTC systems offer a growing range of optimization-related features, either as built-in functions or as add-on products. The modeling environments listed above are aimed at meeting the needs of different types of users. User categories include educational (instructors and students); research scientists, engineers, consultants, and other practitioners (possibly, but not necessarily equipped with an in-depth optimization-related background); optimization experts, software application developers, and other “power users.” (Observe that the user categories listed are not necessarily disjoint.) The pros and cons of the individual software products–in terms

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of their hardware and software demands, ease of usage, model prototyping options, detailed code development and maintenance features, optimization model checking and processing tools, availability of solver options and other auxiliary tools, program execution speed, overall level of system integration, quality of related documentation and support, customization options, and communication with end users–make the corresponding modeling and solver approaches more or less attractive for the various user groups. Given the almost overwhelming amount of topical information, in short, which are the currently available platform and solver engine choices for the GO researcher or practitioner? The more than a decade-old software review (Pintér, 1996b; also available at the Web site of Mittelmann, 2006) listed a few dozen individual software products, including several Web sites with further software collections. Neumaier’s (2006) Web page currently lists more than 100 software development projects. Both of these Web sites include generalpurpose solvers, as well as application-specific products. (It is noted that quite a few of the links in these software listings are now obsolete, or have been changed.) The user’s preference obviously depends on many factors. A key question is whether one prefers to use “free” (noncommercial, research, or even open source) code, or looks for a “ready-to-use” professionally supported commercial product. There is a significant body of freely available solvers, although the quality of solvers and their documentation arguably varies. (Of course, this remark could well apply also to commercial products.) Instead of trying to impose personal judgment on any of the products mentioned in this work, the reader is encouraged to do some Web browsing and experimentation, as his or her time and resources allow. Both Mittelmann (2006) and Neumaier (2006) provide more extensive information on noncommercial, as opposed to commercial, systems. Here we mention several software products that are part of commercial systems, typically as an add-on option, but in some cases as a built-in option. Needless to say, although this author (being also a professional software developer) may have opinions, the alphabetical listing presented below is strictly matter-of-fact. We list only currently available products that are explicitly targeted towards global optimization, as advertised by the Web sites of the listed companies. For this reason, nonlinear (local) solvers are, as a rule, not listed here; furthermore, we do not list modeling environments that currently have no global solver options. AIMMS, by Paragon Decision Technology (www.aimms.com). The BARON and LGO global solver engines are offered with this modeling system as addon options. Excel Premium Solver Platform (PSP), by Frontline Systems (www.solver .com): The developers of the PSP offer a global presolver option to be used with several of their local optimization engines: these currently include LSGRG, LSSQP, and KNITRO. Frontline Systems also offers (as


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genuine global solvers) an Interval Global Solver, an Evolutionary Solver, and OptQuest. GAMS, by the GAMS Development Corporation (www.gams.com). Currently, BARON, DICOPT, LGO, MSNLP, OQNLP, and SBB are offered as solver options for global optimization. LINDO, by LINDO Systems (www.lindo.com). Both the LINGO modeling environment and What’sBest! (the company’s spreadsheet solver) have builtin global solver functionality. Maple, by Maplesoft (www.maplesoft.com) offers the Global Optimization Toolbox as an add-on product. Mathematica, by Wolfram Research (www.wolfram.com) has a built-in function (called NMinimize) for numerical global optimization. In addition, there are several third-party GO packages that can be directly linked to Mathematica: these are Global Optimization, MathOptimizer, and MathOptimizer Professional. MPL, by Maximal Software (www.maximal-usa.com). The LGO solver engine is offered as an add-on. TOMLAB, by TOMLAB Optimization AB (www.tomopt.com) is an optimization platform for solving MATLAB models. The TOMLAB global solvers include CGO, LGO, MINLP, and OQNLP. Note that MATLAB’s own Genetic Algorithm and Direct Search Toolboxes also have heuristic global solver capabilities. To illustrate the functionality and usage of global optimization software, next we review the key features of the LGO solver engine, and then apply its Maple platform-specific implementation in several numerical examples.

11.4 The LGO Solver Suite and Its Implementations 11.4.1 LGO: Key Features The Lipschitz Global Optimizer (LGO) solver suite has been developed and used for more than a decade. The top-level design of LGO is based on the seamless combination of theoretically convergent global and efficient local search strategies. Currently, LGO offers the following solver options. • Adaptive partition and search (branch-and-bound) based global search (BB) • Adaptive global random search (single-start) (GARS) • Adaptive global random search (multistart) (MS) • Constrained local search by the generalized reduced gradient (GRG) method (LS). In a typical LGO optimization run, the user selects one of the global (BB, GARS, MS) solver options; this search phase is then automatically followed

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by the LS option. It is also possible to apply only the LS solver option, making use of an automatically set (default) or a user-supplied initial solution. The global search methodology implemented in LGO is based on the detailed exposition in Pintér (1996a), with many added numerical features. The well-known GRG method is discussed in numerous articles and textbooks; consult for instance Edgar et al. (2001). Therefore only a very brief overview of the LGO component algorithms is provided here. BB, GARS, and MS are all based on globally convergent search methods. Specifically, in Lipschitz-continuous models with suitable Lipschitz-constant (over)estimates for all model functions BB theoretically generates a sequence of search points that will converge to the global solution point. If there is a countable set of such optimal points, then a convergent search point sequence will be generated in association with each of these. In a GO model with a continuous structure (but without postulating access to Lipschitz information), both GARS and MS are globally convergent, with probability one (w.p. 1). In other words, the sequence of points that is associated with the generated sequence of global optimum estimates will converge to a point which belongs to X ∗ , with probability one. (Again, if several such convergent point sequences are generated by the stochastic search procedure, then each of these sequences has a corresponding limit point in X ∗ , w.p. 1.) The LS method (GRG) is aimed at finding a locally optimal solution that satisfies the Karush—Kuhn—Tucker system of necessary local optimality conditions, assuming standard model smoothness and regularity conditions. In all three global search modes the model functions are aggregated by an exact penalty (merit) function. By contrast, in the local search phase all model functions are considered and handled individually. The global search phases incorporate both deterministic and stochastic sampling procedures: the latter support the usage of statistical bound estimation methods, under basic continuity assumptions. All LGO component algorithms are derivativefree. In the global search phase, BB, GARS, and MS use only direct sampling information based on generated points and corresponding model function values. In the LS phase central differences are used to approximate function gradients (under a postulated locally smooth model structure). This direct search approach reflects our objective to handle also models defined by merely computable, continuous functions, including completely “black box” systems. In numerical practice–with finite runs, and user-defined or default option settings–the LGO global solver options generate a global solution estimate that is subsequently refined by the local search mode. If the LS mode is used without a preceding global search phase, then LGO serves as a generalpurpose local solver engine. The expected practical outcome of using LGO to solve a model (barring numerical problems which could impede any numerical method) is a global-search-based feasible solution that meets at least the local optimality conditions. Extensive numerical tests and a range of practical applications demonstrate that LGO can locate the global solution not only


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in the usual academic test problems, but also in more complicated, sizeable GO models: this point is illustrated later on in Sections 11.5 and 11.6. (At the same time, keep in mind the caveats mentioned earlier regarding the performance of any global solver: nothing will “always” work satisfactorily, under resource limitations.)

11.4.2 LGO Implementations The current platform-specific implementations include the following. • LGO with a text input/output interface, for C and FORTRAN compiler platforms • LGO integrated development environment with a Microsoft Windows style menu interface, for C and FORTRAN compiler platforms • AIMMS /LGO solver engine • AMPL /LGO solver engine • GAMS /LGO solver engine • Global Optimization Toolbox for Maple (the LGO solver linked to Maple as a callable add-on package) • MathOptimizer Professional, with an LGO solver engine link to Mathematica • MPL /LGO solver engine • TOMLAB /LGO, for MATLAB users Technical descriptions of these software implementations, including detailed numerical tests and a range of applications, have appeared elsewhere. For implementation details and illustrative results, consult Pintér (1996a, 1997, 2001a,b, 2002a,b, 2003b, 2005), as well as Pintér and Kampas (2003) and Pintér et al. (2004, 2006). The compiler-based LGO solver suite can be used in standalone mode, and also as a solver option in various modeling environments. In its core (text input/output based) implementation version, LGO reads an input text file that contains application-specific (model descriptor) information, as well as a few key solver options (global solver type, precision settings, resource and time limits). During the program run, LGO makes calls to an application-specific model function file that returns function values for the algorithmically chosen sequence of arguments. Upon completing the LGO run, automatically generated summary and detailed report files are available. As can be expected, this LGO version has the lowest demands for hardware; it also runs fastest, and it can be directly embedded into various decision support systems, including proprietary user applications. The same core LGO system is also available in directly callable form, without reading and writing text file: this version is frequently used as a built-in solver module in other (general-purpose or customized modeling) systems.

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LGO can also be equipped, as a readily available (implemented) option, with a Microsoft Windows style menu interface. This enhanced version is referred to as the LGO Integrated Development Environment (IDE). The LGO IDE supports model development, compilation, linking, execution, and the inspection of results, together with built-in basic help facilities. In the two LGO implementations mentioned above, models can be connected to LGO using one of several programming languages that are available on personal computers and workstations. Currently supported platforms include, in principle, “all” professional FORTRAN 77/90/95 and C/C++ compilers. Examples of supported compilers include Compag, Intel, Lahey, and Salford FORTRAN, as well as g77 and g95, and Borland and Microsoft C/C++. Other customized versions (to use with other compilers or software applications) can also be made available upon request. In the optimization modeling language (AIMMS, AMPL, GAMS, and MPL) or ISTC (Maple, Mathematica, and TOMLAB) environments the core LGO solver engine is seamlessly linked to the corresponding modeling platform, as a dynamically callable or shared library, or as an executable program. The key advantage of using LGO within a modeling or ISTC environment is the combination of modeling-system-specific features, such as model prototyping and detailed development, model consistency checking, integrated documentation, visualization, and other platform-specific features, with a numerical performance comparable to that of the standalone LGO solver suite. For peer reviews of several of the listed implementations, the reader is referred to Benson and Sun (2000) on the core LGO solver suite, Cogan (2003) on MathOptimizer Professional, and Castillo (2005), Henrion (2006), and Wass (2006) on the Global Optimization Toolbox for Maple. Let us also mention here that LGO serves to illustrate global optimization software (in connection with a demo version of the MPL modeling system) in the prominent O.R. textbook by Hillier and Lieberman (2005).

11.5 Illustrative Examples In order to present some small-scale, yet nontrivial numerical examples, in this section we illustrate the functionality of the LGO software as it is implemented in the Global Optimization Toolbox (GOT) for Maple. Maple (Maplesoft, 2006) enables the development of interactive documents called worksheets. Maple worksheets can incorporate technical model description, combined with computing, programming, and visualization features. Maple includes several thousands of built-in (directly callable) functions to support the modeling and computational needs of scientists and engineers. Maple also offers a detailed online help and documentation system with ready-to-use examples, topical tutorials, manuals, and Web links, as well as a built-in mathematical dictionary. Application development is assisted by


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debugging tools, and automated (ANSI C, FORTRAN 77, Java, Visual Basic, and MATLAB) code generation. Document production features include HTML, MathML, TeX, and RTF converters. These capabilities accelerate and expand the scope of the optimization model development and solution process. Maple, similarly to other modeling environments, is portable across all major hardware platforms and operating systems (including Windows, Macintosh, Linux, and UNIX versions). Without going into further details on Maple itself, we refer to the Web site www.maplesoft.com that offers in-depth topical information, including product demos and downloadable technical materials. The core of the Global Optimization Toolbox for Maple is a customized implementation of the LGO solver suite (Maplesoft, 2004) that, as an add-on product, upon installation, can be fully integrated with Maple. The advantage of this approach is that, in principle, the GOT can readily handle “all” continuous model functions that can be defined in Maple, including also new (user-defined) functions. We do not wish to go into programming details here, and assume that the key ideas shown by the illustrative Maple code snippets are easily understandable to all readers with some programming experience. Maple commands are typeset in Courier bold font, following the so-called classic Maple input format. The input commands are typically followed by Maple output lines, unless the latter are suppressed by using the symbol “:” instead of “;” at the end of an input line. In the numerical experiments described below, an AMD Athlon 64 (3200+, 2GHz) processor-based desktop computer has been used that runs under Windows XP Professional (Version 2002, Service Pack 2).

11.5.1 Getting Started with the Global Optimization Toolbox To illustrate the basic usage of the Toolbox, let us revisit model (11.3). The Maple command > with(GlobalOptimization); makes possible the direct invocation of the subsequently issued, GOT related, commands. Then the next Maple command numerically solves model (11.3): the response line below the command displays the approximate optimum value, and the corresponding solution argument. > GlobalSolve(cos(x)*sin(x^2-x), x=1..10); [—.990613849411236758, [x = 9.28788130421885682]]

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The detailed runtime information not shown here indicates that the total number of function evaluations is 1262; the associated runtime is a small fraction of a second. Recall here Figure 11.1 which, after careful inspection, indicates that this is indeed the (approximate) global solution. (One can also see that the default visualization–similarly to other modeling environments–has some difficulties to depict this rapidly changing function.) There are several local solutions that are fairly close to the global one: two of these numerical solutions are [—.979663995439954860, [x = 3.34051270473064265]], and [—.969554320487729716, [x = 6.52971402762202757]]. Similarly, the next statement returns an approximate global solution in the visibly nontrivial model (11.4): > GlobalSolve(cos(x)*sin(y^2-x)+cos(y)*sin(x^2-y), x=1..10, y=1..10); [—1.95734692335253380, [x = 3.27384194476651214, y = 6.02334184076140478]]. The result shown above has been obtained using GOT default settings: the total number of function evaluations in this case is 2587, and the runtime is still practically zero. Recall now also Figure 11.2 and the discussion related to the possibly numerical difficulty of GO models. The solution found by the GOT is global-search-based, but without a rigorous deterministic guarantee of its quality. Let us emphasize that to obtain such guarantees (e.g., by using interval-arithmetic-based solution techniques) can be a very resourcedemanding exercise, especially in more complex and/or higher-dimensional models, and that it may not be possible, for example, in “black box” situations. A straightforward way to attempt finding a better quality solution is to increase the allocated global search effort. Theoretically, using an “infinite” global search effort will lead to an arbitrarily close numerical estimate of the global optimum value. In the next statement we set the global search effort to 1000000 steps (this limit is applied only approximately, due to the possible activation of other stopping criteria): > GlobalSolve(cos(x)*sin(y^2-x)+cos(y)*sin(x^2-y), x=1..10, y=1..10, evaluationlimit=1000000, noimprovementlimit=1000000);


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[—1.98122769882222882, [x = 9.28788128193757068, y = 9.28788127177065270]]. Evidently, we have found an improved solution, at the expense of a significantly increased global search effort. (Now the total number of function evaluations is 942439, and the runtime is approximately 5 seconds.) In general, more search effort can always be added, in order to verify or perhaps improve the incumbent numerical solution. Comparing now the solution obtained to that of model (11.3), and observing the obvious formal connection between the two models, one can deduce that now we have found a close numerical approximation of the true global solution. Simple modeling insight also tells us that the global solution in model (11.4) is bounded from below by —2. Hence, even without Figures 11.1 and 11.2 we would know that the solution estimates produced above must be fairly close to the best possible solution. The presented examples illustrate several important points. • Global optimization models can be truly difficult to solve numerically, even in (very) low dimensions. • It is not always possible to “guess” the level of difficulty. One cannot always (or at all) generate model visualizations similar to Figures 11.1 and 11.2, even in chosen variable subspaces, because it could be too expensive numerically, even if we have access to suitable graphics facilities. Insight and model-specific expertise can help significantly, and these should be used whenever possible. • There is no solver that will handle all possible instances from the general CGO model class within an arbitrary prefixed amount of search effort. In practice, one needs to select and recommend default solver parameters and options that “work well in most cases, based on an acceptable amount of effort.” Considering the fact that practically motivated modeling studies are often supported only by noisy and/or scarce data, this pragmatic approach is justifiable in many practical situations. • The default solver settings should return a global-search-based highquality feasible solution (arguably, the models (11.3) and (11.4) can be considered as difficult instances for their low dimensionality). Furthermore, it should be easy to modify the default solver settings and to repeat runs, if this is deemed necessary. The GOT software implementation automatically sets default parameter values for its operations, partly based on the model to solve. These settings are suitable in most cases, but the user can always assign (i.e., override) them. Specifically, one can select the following options and parameter values. • Minimization or maximization model • Search method (BB+LS, GARS+LS, MS+LS, or standalone LS)

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• Initial solution vector setting (used by the LS operational mode), if available • Constraint penalty multiplier: this is used by BB, GARS, and MS, in an aggregated merit function (recall that the LS method handles all model functions individually) • Maximal number of merit function evaluations in the selected global search mode • Maximal number of merit function evaluations in the global search mode, without merit function value improvement • Acceptable target value for the merit function, to trigger an “operational switch” from global to local search mode • Feasibility tolerance used in LS mode • Karush—Kuhn—Tucker local optimality tolerance in LS mode • Solution (computation) time limit For further information regarding the GOT, consult the product Web page (Maplesoft, 2004), the article (Pintér et al., 2006), and the related Maple help system entries. The product page also includes links to detailed interactive demos, as well as to downloadable application examples.

11.5.2 Handling (General) Constrained Global Optimization Models Systems of nonlinear equations play a fundamental role in quantitative studies, because equations are often used to characterize the equilibrium states and optimality conditions of physical, chemical, biological, or other systems. In the next example we formulate and solve a system of equations. At the same time, we also illustrate the use of a general model development style that is easy to follow in Maple, and–mutatis mutandis–also in other modeling systems. Consider the equations > eq1 := exp(x-y)+sin(2*x)-cos(y+z)=0: eq2 := 4*x-exp(z-y)+5*sin(6*x-y)+3*cos(3*x*y)=0: eq3 := x*y*z-10=0:

(11.5)

To solve this system of equations, let us define the optimization model components as shown below (notice the dummy objective function). > constraints := eq1,eq2,eq3: > bounds := x=-2..2, y=-1..3, z=2..4: > objective:=0: Then the next Maple command is aimed at generating a numerical solution to (11.5), if such solution exists.


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> solution:= GlobalSolve(objective, constraints, bounds); solution:=[0., [x=1.32345978290539557,y=2.78220763578413344,z=2.71581206431678090]]. This solution satisfies all three equations with less than 10−9 error, as verified by the next statement: > eval(constraints, solution[2]); {−0.1 · 10−9 = 0, −0.6 · 10−9 = 0, 0 = 0} Without going into details, let us note that multiple solutions to (11.5) can be found (if such solutions exist), for example, by iteratively adding constraints that will exclude the solution(s) found previously. Furthermore, if a system of equations has no solutions, then using the GOT we can obtain an approximate solution that has globally minimal error over the box search region, in a given norm: consult Pintér (1996a) for details. Next, we illustrate the usage of the GOT in interactive mode. The statement shown below directly leads to the Global Optimization Assistant dialog, see Figure 11.3. > solution:= Interactive(objective, constraints, bounds); Using the dialog, one can also directly edit (modify) the model formulation if necessary. The figure shows that the default (MS+LS) GOT solver mode returns the solution presented above. Let us point out here that none of the local solver options indicated in the Global Optimization Assistant (see the radio buttons under Solver) is able to find a feasible solution to this model. This finding is not unexpected: rather, it shows the need for a global scope search approach to handle this model and many other similar problems. Following the numerical solution step, one can press the Plot button (shown in the lower right corner in Figure 11.3). This will invoke the Global Optimization Plotter dialog shown in Figure 11.4. In the given subspace (x, y) that can be selected by the GOT user, the surface plot shows the identically zero objective function. Furthermore, on its surface level one can see the constraint curves and the location of the global solution found: in the original color figure this is a light green dot close to the boundary as indicated by the numerical values found above. Notice also the option to select alternative subspaces (defined by variable pairs) for visualization. The figures can be rotated, thereby offering the possibility of detailed model function inspection. Such inspection can help users to increase their understanding of the model.

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Fig. 11.3 Global Optimization Assistant dialog for model (11.5).

Fig. 11.4 Global Optimization Plotter dialog for model (11.5).



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11.5.3 Optimization Models with Embedded Computable Functions It was pointed out earlier (in Section 11.1) that in advanced decision models some model functions may require the execution of various computational procedures. One of the advantages of using an ISTC system such as Maple is that the needed functionality to perform these operations is often readily available, or directly programmable. To illustrate this point, in the next example we show the globally optimized argument value of an objective function defined by Bessel functions. As it is known, the function BesselJ(ν, x) satisfies Bessel’s differential equation x2 y 00 + xy 0 + (x2 − ν 2 )y = 0.

(11.6)

In (11.6) x is the function argument, and the real value ν is the order (or index parameter) of the function. The evaluation of BesselJ requires the solution function of the differential equation (11.6), for the given value of ν, and then the calculation of the corresponding function value for argument x. For example, BesselJ(0, 2)∼0.2238907791; consult Maple’s help system for further details. Consider now the optimization model defined and solved below: > objective:=BesselJ(2,x)*BesselJ(3,y)BesselJ(5,y)*BesselJ(7,x): > bounds := x=-10..20, y=-15..10: > solution:=GlobalSolve(objective, bounds);

(11.7)

solution := [—.211783151218360000, [x = —3.06210564091438720, y = —4.20467390983796196]]. The corresponding external solver runtime is about 4 seconds. The next figure visualizes the box-constrained optimization model (11.7). Here a simple inspection and rotation of Figure 11.5 helps to verify that the global solution is found indeed. Of course, this would not be directly possible in general (higher-dimensional or more complicated) models: recall the related earlier discussion and recommendations from Section 11.5.1.

11.6 Global Optimization: Applications and Perspectives In recent decades, global optimization gradually has become an established discipline that is now taught worldwide at leading academic institutions. GO methods and software are also increasingly applied in various research contexts, including industrial and consulting practice. The currently available

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Fig. 11.5 Optimization model objective defined by Bessel functions.

professional software implementations are routinely used to solve models with tens, hundreds, and sometimes even thousands of variables and constraints. Recall again the caveats mentioned earlier regarding the potential numerical difficulty of model instances: if one is interested in a guaranteed high-quality solution, then the necessary runtimes could become hours (or days, or more), even on today’s high-performance computers. One can expect further speedup due to both algorithmic improvements and progress in hardware/software technology, but the theoretically exponential “curse of dimensionality” associated with the subject of GO will always be there. In the most general terms, global optimization technology is well suited to analyze and solve models in advanced (acoustic, aerospace, chemical, control, electrical, environmental, and other) engineering, biotechnology, econometrics and financial modeling, medical and pharmaceutical studies, process industries, telecommunications, and other areas. For detailed discussions of examples and case studies consult, for example, Grossmann (1996), Pardalos et al. (1996), Pintér (1996a), Corliss and Kearfott (1999), Papalambros and Wilde (2000), Edgar et al. (2001), Gao et al. (2001), Schittkowski (2002), Tawarmalani and Sahinidis (2002), Zabinsky (2003), Neumaier (2006), Nowak (2005), and Pintér (2006a), as well as other topical works. For example, recent numerical studies and applications in which LGO implementations have been used are described in the following works: • Cancer therapy planning (Tervo et al., 2003)


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• Combined finite element modeling and optimization in sonar equipment design (Pintér and Purcell, 2003) • Laser equipment design (Isenor et al., 2003) • Model calibration (Pintér, 2003a, 2006b) • Numerical performance analysis on a collection of test and “real-world” models (Pintér, 2003b, 2006b) • Physical object configuration analysis and design (Kampas and Pintér, 2006) • Potential energy models in computational chemistry (Pintér, 2000, 2001b, Stortelder et al., 2001) • Circle packing models and their industrial applications (Kampas and Pintér, 2004, Pintér and Kampas, 2005a,b, Castillo et al., 2008) The forthcoming volumes by Kampas and Pintér (2009) and Pintér (2009) also discuss a large variety of GO applications, with extensive references.

11.7 Conclusions Global optimization is a subject of growing practical interest as indicated by recent software implementations and by an increasing range of applications. In this work we have discussed some of these developments, with an emphasis on practical aspects. In spite of remarkable progress, global optimization remains a field of extreme numerical challenges, not only when considering “all possible” GO models, but also in practical attempts to handle complex and sizeable problems within an acceptable timeframe. The present discussion advocates a practical solution approach that combines theoretically rigorous global search strategies with efficient local search methodology, in integrated, flexible solver suites. The illustrative examples presented here, as well as the applications referred to above, indicate the practical viability of such an approach. The practice of global optimization is expected to grow dynamically. We welcome feedback regarding current and future development directions, new test challenges, and new application areas. Acknowledgments First of all, I wish to thank David Gao and Hanif Sherali for their kind invitation to the CDGO 2005 conference (Blacksburg, VA, August 2005), as well as for the invitation to contribute to the present volume dedicated to Gilbert Strang on the occasion of his 70th birthday. Thanks are due to an anonymous reviewer for his/her careful reading of the manuscript, and for the suggested corrections and modifications. I also wish to thank my past and present developer partners and colleagues–including AMPL LLC, Frontline Systems, the GAMS Development Corporation, Frank Kampas, Lahey Computer Systems, LINDO Systems, Maplesoft, Mathsoft, Maximal Software, Paragon Decision Technology, The Mathworks, TOMLAB AB, and Wolfram Research–for cooperation, quality software and related documentation, and technical support.

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In addition to professional contributions and in-kind support offered by developer partners, the research work summarized and reviewed in this chapter has received partial financial support in recent years from the following organizations: DRDC Atlantic Region, Canada (Contract W7707-01-0746), the Dutch Technology Foundation (STW Grant CWI55.3638), the Hungarian Scientific Research Fund (OTKA Grant T 034350), Maplesoft, the National Research Council of Canada (NRC IRAP Project 362093), the University of Kuopio, and Wolfram Research. Special thanks are due to our growing clientele, and to all reviewers and testers of our various software implementations, for valuable feedback, comments, and suggestions.

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Chapter 12

Two-Stage Stochastic Mixed-Integer Programs: Algorithms and Insights Hanif D. Sherali and Xiaomei Zhu

Summary. Stochastic (mixed-) integer programs pose a great algorithmic and computational challenge in that they combine two generally difficult classes of problems: stochastic programs and discrete optimization problems. Exploring its dual angular structure, various decomposition methods have been widely studied, including Benders’ decomposition, Lagrangian relaxation, and test-set decomposition. These decomposition methods are often combined with search procedures such as branch-and-bound or branch-andcut. Within the confines of these broad frameworks, fine-tuned algorithms have been proposed to overcome obstacles such as nonconvexity of the secondstage value functions under integer recourse, and to take advantage of the many similar structured scenario subproblems using variable transformations. In this chapter, we survey some recent algorithms developed to solve twostage stochastic (mixed-) integer programs, as well as provide insights into and results concerning their interconnections, particularly for alternative convexification techniques. Key words: Two-stage stochastic mixed-integer programs, L-shaped method, Benders’ decomposition, branch-and-cut, convexification, reformulationlinearization technique (RLT), disjunctive programming

12.1 Introduction In this chapter, we discuss two-stage stochastic mixed-integer programs (SMIP) of the following form.

Hanif D. Sherali · Xiaomei Zhu Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA 24061, U.S.A., e-mail: [email protected], [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_12, © Springer Science+Business Media, LLC 2009

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H.D. Sherali, X. Zhu

SMIP:

Minimize cx + E[f (x, ω ˜ )]

subject to Ax ≥ b

(12.1a) (12.1b)

x≥0

(12.1c)

xi binary, ∀i ∈ Ib ⊆ I ≡ {1, ..., n1 }

(12.1d)

xi integer, ∀i ∈ Iint ⊆ I \ Ib ,

(12.1e)

˜ A, ˜ P) ˜ (with where ω ˜ is a random variable defined on a probability space (Ω, ˜ A, ˜ and P, ˜ respectively, denoting the set of all outcomes, a collection of Ω, random variables, and their associated probability distributions), and where for any given realization ω of ω ˜ , we have f (x, ω) = minimum g(ω)y subject to W (ω)y ≥ r(ω) − T (ω)x

(12.1f) (12.1g)

y≥0

(12.1h)

yj binary, ∀j ∈ Jb ⊆ J ≡ {1, ..., n2 }

(12.1i)

yj integer, ∀j ∈ Jint ⊆ J \ Jb .

(12.1j)

In the above, A is an m1 × n1 matrix, and for each ω, W (ω) is an m2 × n2 recourse matrix, T (ω) is an m2 × n1 technology matrix, and the other defined vectors are of corresponding conformable sizes. We assume that the elements of A, T (ω), and W (ω) are rational (so that they are scalable to integers, if necessary). For computational viability, a finite number of scenarios, denoted as a set S and indexed by s, is often considered based on some discretization of the possible realizations of ω ˜ , each with an associated probability of occurrence ps , s ∈ S. (See [29] for a justification on approximating continuously distributed scenario parameters by a discrete distribution having a finite support.) Accordingly, the realizations of g(ω), W (ω), T (ω), and r(ω) are correspondingly denoted as gs , Ws , Ts , and rs , respectively, for s ∈ S. In this chapter, we assume such a discrete probability distribution. Arguably, SMIP is among the most challenging of optimization problems because it combines two generally difficult classes of problems: stochastic programs and discrete optimization problems. Extended from theories particularly in large scale optimization and integer programming ([25, 24]), researchers have actively studied the properties and solution approaches for such problems. We refer the reader to the extensive survey papers by Schultz et al. [31], Klein Haneveld and van der Vlerk [21], Schultz [30], and Sen [34], and an annotated bibliography by Stougie and van der Vlerk [42] for a discussion on the principal properties of SMIPs and some earlier algorithmic developments in this area. A group of important stochastic integer programs are problems having simple integer recourse. Research in this area, such as

12 Two-Stage Stochastic Mixed-Integer Programs

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by Klein Haneveld et al. [19, 20], have been included in detail in various aforementioned surveys. In this chapter, we focus on certain recent algorithmic advances in solving two-stage stochastic (mixed-) integer programs. We also provide insights into their interconnections, and present some results relating the convexification strategies afforded by the reformulation-linearization technique (RLT) and disjunctive programming methods for solving mixed-integer 0—1 SMIPs. The remainder of this chapter is organized as follows. In Section 12.2, we survey different decomposition frameworks that have been used to solve SMIPs, including decomposition by stage (primal decomposition), decomposition by scenario (dual decomposition), and test-set decomposition. In Section 12.3, we exhibit certain insightful relationships between some convexification methods that have been used for solving mixed-integer 0—1 stochastic problems, particularly, employing disjunctive programming and RLT-based cutting plane methods. In Section 12.4, we discuss three enumerative methods using tender variables when the technology matrix is fixed. We conclude this chapter in Section 12.5.

12.2 Stage, Scenario, and Test-Set Decomposition Collecting the problems for the two stages together, and denoting the constraints (12.1b)—(12.1e) as x ∈ X and the constraints (12.1h)—(12.1j) written for scenario s as y s ∈ Y s , the deterministic equivalent form for (12.1) is given as follows. X Minimize cx + ps gs y s (12.2a) s∈S

subject to Ts x + Ws y s = rs , x ∈ X, y s ∈ Y s ,

∀s ∈ S

(12.2b)

∀s ∈ S.

(12.2c)

This representation reveals a dual angular structure that lends itself well to decomposition schemes. We discuss two major groups of decomposition methods, by stages and by scenarios, as well as a novel approach called testset decomposition. Decomposition by stages, also known as primal decomposition, essentially adopts the framework of Benders’ decomposition (Benders [9]), and is more popularly known as the L-shaped method (Van Slyke and Wets [43]) in the context of stochastic programming. In this approach, for each first-stage solution x ¯ produced by a master program, one solves the corresponding second-stage problem (also called a scenario subproblem, or simply subproblem) based on Problem (12.2) with x fixed at x ¯. We discuss these methods in Section 12.2.1. Scenario decomposition methods, also known as dual decomposition, work with relaxing and restoring the nonanticipativ-

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ity condition, which, in a two-stage problem setting, simply means that all scenario outcomes should be based on some identical first-stage solution. When this condition is relaxed, smaller-scaled problems corresponding to scenarios are obtained that are easier to solve, but they also yield different first-stage solutions, which need to be reconciled. These methods are covered in Section 12.2.2. In Section 12.2.3, we describe the test-set decomposition approach by Hemmecke and Schultz [17], which decomposes the problem’s Graver test-sets, instead of the problem itself.

12.2.1 Stage Decomposition — L-Shaped Methods The simplest form of stochastic mixed-integer programs contains purely binary first-stage variables and purely continuous second-stage variables. In this case, Benders’ decomposition, or the L-shaped method, can be easily applied using some form of enumeration on the binary first-stage variables, as in Wollmer [44]. Alternatively, and also more generally, if the secondstage problems are easy to solve (not necessarily purely continuous), then these problems can be solved using the integer L-shaped method of Laporte and Louveaux [23]. This is a branch-and-cut (B&C) procedure that is implemented in the projected space of the first-stage variables. At each node, the two-stage problem is solved using Benders’ decomposition, which generates feasibility and/or optimality cuts for the first-stage solution, as necessary. The Benders master problem (or the “current problem” of the L-shaped method) at some node q then takes the form: Minimize cx + η subject to Dk x ≥ dk , Lk x ≥ lk ,

(12.3a) ∀k = 1, . . . , Fq

(12.3b)

∀k = 1, . . . , Oq

(12.3c)

x ∈ X, η unrestricted,

(12.3d)

where k = 1, . . . , Fq and k = 1, . . . , Oq , respectively, index the feasibility cuts (12.3b) and the optimality cuts (12.3c) at node q, and η is a variable representing an approximation of the expected second-stage objective value (or the second-stage value function). If feasibility of the subproblems is guaranteed for any fixed x, or x ∈ X, respectively known as the case of complete recourse or relatively complete recourse, then feasibility cuts will not be generated from the second-stage problems, and (12.3b) will only include certain relevant constraints that fix particular components of x at 0 or 1 as per the branching restrictions. A nodal problem at node q in the B&C tree is resolved whenever a feasibility or optimality cut is added. It is fathomed by the infeasibility of (12.3) or by the bounding processes, and is otherwise par-


409

titioned via branching if the solution to the continuous relaxation does not satisfy the binary restrictions. Unlike in a typical branch-and-bound (B&B) process, whenever integrality is satisfied and reveals a potentially better incumbent solution to the current (relaxed) master program, the node is not necessarily fathomed. Instead, the subproblem is solved to possibly generate an optimality cut. Theoretically, this framework of embedding the L-shaped method in a B&C process applies to two-stage SMIPs in which the first stage contains binary or integer variables and valid optimality cuts are obtainable from the second-stage problems. In practice, however, valid optimality cuts are not easy to derive unless the second stage contains purely continuous variables, or the first stage contains purely binary variables. In the former case, the second-stage value functions fs (·) are piecewise linear and convex, and so, valid optimality cuts can be derived using the dual variables directly. In the latter case, for any binary feasible solution xr , defining Ir ≡ {i ∈ I : x ¯ri = 1}, the following is a set of valid optimality cuts. X £X ¤ η ≥ (Q(xr ) − L) xi − xi − |Ir | + 1 + L, ∀r = 1, . . . , R, (12.4) i∈Ir

P

i∈I / r

where Q(x) ≡ s ps fs (x), L is a lower bound on Q, and 1, . . . , R are indices of all the binary feasible solutions that have been encountered thus far. When the second-stage problems contain integer variables, their value functions fs (·) are lower semicontinuous (Blair and Jeroslow [11]) and in general nonconvex and discontinuous. Thus optimality cuts are not readily available from dual variables as in the continuous case. In such cases, when the two-stage program has a deterministic cost vector g and recourse matrix W , Carøe and Tind [14] propose to use a subset F of the dual price functions of the second-stage integer programs (see Nemhauser and Wolsey [25] for dual price functions for integer programs) to derive feasibility cuts and optimality cuts. These functions are, however, nonlinear in general, resulting in a nonlinear form of Problem (12.3). Moreover, the dual price function class F has to be sufficiently large so that the duality gap between the second-stage problem and its (F-) dual is closed. The authors show that this is achieved when the second-stage problems are solved using cutting plane techniques or branch-and-bound. In these cases, finite termination is also established, given that (12.3) can be solved finitely. In particular, when Gomory’s cuts are applied, the dual price functions can be transformed into linear functions having mixed-integer variables, and (12.3) is then a mixed-integer linear problem in lieu of a nonlinear problem. The above decomposition methods work with the primal second-stage problems, given any fixed first-stage solution. Within this framework, alternative methods have been developed to obtain optimality cuts in the case of integer recourse. We discuss some of these methods in Section 12.3. In Section 12.2.2 below, we turn to another type of decomposition approach

410


that relaxes the nonanticipativity condition and works with copies of the first-stage solution to restore this condition.

12.2.2 Scenario Decomposition — Relaxation of the Nonanticipativity Condition The second-stage problems are related to each other through the first-stage solution only. If we make a copy of the first-stage variable x for each scenario, say xs for s ∈ S, and enforce condition, x1 = xs , ∀s ∈ S, P the nonanticipativity s s s conveniently written as s∈S H x = 0 for some suitable Hl×n , then the 1 deterministic equivalent problem shown in (12.2) can be rewritten as follows, X ©X ª min ps (cxs + gs y s ) : (12.2b), (12.2c), H s xs = 0 . (12.5) s∈S

s∈S

ª ©P The Lagrangian dual of (12.5) is max s∈S Ds (λ) , where λ∈Rl

Ds (λ) ≡ min{ps (cxs + gs y s ) + λ(H s xs ) : (12.2b), (12.2c)}, ∀s ∈ S.

The Lagrangian dual value yields a tighter lower bound for Problem (12.2) than its LP-relaxation. Carøe and Schultz [12] accordingly propose a dual decomposition algorithm that uses the values of the Lagrangian dual as lower bounds in a B&B process. The tradeoff here is having to solve a nonsmooth Lagrangian dual problem compared to a linear program. At each nodal problem, after solving the associated Lagrangian dual problem, the solutions xs , ∀s ∈ S, are averaged and rounded to obtain some x ¯ satisfying the integrality restrictions. The objective value of (12.2) obtained after fixing x at x ¯ is used to update the upper bound. At the branching step, for a selected branching variable xi (component of x), constraints xi ≤ b¯ xi c and xi ≥ d¯ xi e, or xi ≤ x ¯i − and xi ≥ x ¯i + for some tolerance > 0, are applied at the two child nodes, respectively, depending on whether xi is an integer or continuous variable. This algorithm is finitely convergent if X is bounded and the x variables are purely integer-restricted. Schultz and Tiedemann [33] extend this approach to solve stochastic programs that include an additional objective term based on the probability of a risk function exceeding a prespecified threshold value. A survey for SMIPs having risk measures can be found in Schultz [30]. Alonso-Ayuso et al. [3] also relax the nonanticipativity condition in their branch-and-fix coordination algorithm (BFC) for pure 0—1 multistage stochastic programs, or mixed 0—1 two-stage stochastic programs having purely binary variables in the first stage. They assume deterministic technology and recourse matrices. In this algorithm, both the nonanticipativity condition and the binary restrictions on x are relaxed, resulting in a linear problem


411

for each scenario. A B&B tree is developed for each scenario at a terminal node of the scenario tree, and the branch and fix operations are performed on the so-called “twin node families.” In a two-stage setting, a twin node family includes the nodal problems in the B&B trees for all scenarios that have the already fixed x-variables equal to a same value (0 or 1). To enforce the nonanticipativity condition, the same branching variable is selected for the active nodes in a twin node family. Note that these nodes belong to different scenarios. Two new twin node families are then formed by fixing the selected branching variable at 0 and 1, respectively. Lower bounds on the objective value are updated by calculating the deterioration of the objective value due to the fixing. Alonso-Ayuso et al. [4] demonstrated this approach in a set of supply chain planning problems, having up to 3933 constraints and 3768 variables (including 114 binary variables), using an 800 MHz Pentium III Processor with 512 Mb of RAM. However, no comparison with other algorithms or commercial software packages was made.

12.2.3 Decomposition of Test-Sets When the cost vector g, the technology matrix T , and the recourse matrix W are all deterministic, and randomness only appears in the right-hand-side values rs , Hemmecke and Schultz [17] take advantage of the dual angular structure in (12.2) to develop a test-set decomposition-based approach for two-stage stochastic integer programs. If the problem contains continuous variables, certain inherited computational difficulties from such an approach demand extra care. A finite universal test-set for a family of integer problems (IP )c,b :

min{cz : Az = b, z ∈ Zd+ },

(12.6)

where A ∈ Ql×d is fixed, and where c ∈ Rd and b ∈ Rl vary, contains a set of vectors that can be used to solve any problem in this family (IP )c,b for a given c and b (Hemmecke [16]). In this process, an initial feasible solution is first found using the test-set vectors, then an optimal solution is obtained by searching along improving directions, again using the vectors (called improving vectors) in the test-set. The IP Graver test-set (Graver [15]) is one such finite universal test-set, and can be computed using the kernel of A, ker (A), as shown in Hemmecke [16]. A major drawback of using the test-set approach to solve integer programs is that a large number of vectors need to be stored, even for small-sized problems. Hence, for the typically large-sized stochastic integer programs, directly applying this approach is impractical. Exploring the dual angular structure of (12.2), Hemmecke and Schultz [17] show that Graver test-set vectors for this type of problem can be decomposed into, and then constructed

412


from, a small number of building-block vectors, and the latter can then be used to solve large-scaled stochastic integer programs. Let ⎞ ⎛ A 0 0 ... 0 ⎜T W 0 ... 0 ⎟ µ ¶ ⎟ ⎜ A 0 ⎜T 0 W ... 0 ⎟ A|S| ≡ ⎜ . (12.7) ⎟ and A1 ≡ T W ⎜ .. .. .. . . .. ⎟ ⎝. . . . . ⎠ T 0 0 ... W

Observing that (u, v1 , . . . , v|S| ) ∈ ker (A|S| ) ⇔ (u, v1 ), . . . , (u, v|S| ) ∈ ker (A1 ), Hemmecke and Schultz [17] propose to use the individual vectors u, v1 , . . . , and v|S| as building blocks of the vectors (u, v1 , . . . , v|S| ) in the Graver test-set of A|S| . These building blocks, in lieu of the test-set vectors, are collected into a set H∞ and arranged in pairs (u, Vu ), where Vu is the set of vectors v such that (u, v) ∈ ker (A). The set of building blocks H∞ is shown to be finite, and can be computed using a finite algorithm. Although the computation of H∞ is expensive, because it depends only on A, W , and T , and is independent of |S|, cost coefficients, and right-hand-side values, the proposed approach is insensitive to the number of scenarios once H∞ is obtained. After computing and storing the building blocks, the test-set vectors are constructed during the two steps of obtaining an initial feasible solution and finding improving directions to solve (12.2). The finiteness of H∞ guarantees the finiteness of the proposed algorithm.

12.3 Disjunctive Programming and RLT Cutting Plane Methods for Mixed 0—1 Stochastic Programs In this section, we assume the following. A1. No general integer variable exists in either stage; that is, Iint = Jint = ∅. A2. The continuous variables in both stages are bounded. Moreover, for the purpose of exposition, the continuous variables in the second stage are scaled onto [0,1], with the corresponding bounding restrictions y ≤ 1 being absorbed within (or implied by) (12.1g). A3. For any feasible first-stage variable x ∈ X, the second-stage problems are feasible (relative complete recourse). We study two groups of cutting plane approaches for solving two-stage SMIPs having 0—1 mixed-integer variables, and focus on the idea of sharing cut coefficients among scenarios. The first group is based on disjunctive programming, represented by the research of Carøe and Tind [13], Sen and Higle [35], Sen and Sherali [36], and Ntaimo and Sen [27]. The second group applies the reformulation-linearization technique (RLT) to derive cutting planes, and includes the papers by Sherali and Fraticelli [39] and Sherali


413

and Zhu [41]. Both disjunctive cuts and RLT cuts are convexification cutting planes applied to deal with the binary variables in the second-stage problems. In solving stochastic programs, particularly when the number of scenarios is large, memory space is crucial. If some cut coefficient can be shared between scenarios or iterations, we can gain much in memory savings and algorithmic efficacy. In Section 12.3.1, we therefore focus on the convexification process and cut-sharing properties for solving mixed-integer 0—1 second-stage problems. In Section 12.3.2, we then discuss solution approaches for problems containing continuous first-stage variables, as these problems pose an extra challenge on assuring convergence. In Section 12.3.3, we relate the cuts generated using disjunctive programming and the RLT technique to show their interconnections.

12.3.1 Solving Mixed-Integer 0—1 Second-Stage Problems Given an S index set H and some polyhedral sets Ph , for h ∈ H, a disjunctive set P = h∈H Ph is said to be in disjunctive normal form. Using disjunctive programming, we can characterize the closure of the convex hull of P and generate valid inequalities and even facets for this representation (see, for example, Blair and Jeroslow [10], Balas [5], and Sherali and Shetty [40]). The class of 0—1 mixed-integer programs (MIP) belongs to a special type of disjunctive programs called facial disjunctive programs, written in conjunctive normal form, that is, a conjunction over i ∈ Ib of the disjunction {xi = 0 ∨ xi = 1}. For this type of programs, we can obtain the convex hull of the feasible solution set using a sequential convexification process. The lift-and-project algorithm proposed by Balas et al. [7, 8, 6] solves 0—1 mixed-integer programs using disjunctive programming. Carøe and Tind [13] modified this approach and applied it to the deterministic equivalent form of SMIP (Problem (12.2)) in which the first-stage variables are purely continuous (i.e., Ib = ∅) and the second stage contains both continuous and binary variables. Let P s be the set of solutions in the space of (x, y h , ∀h ∈ S) when only the constraints on x and y s are considered. A direct application of the lift-and-project method would generate cuts in the (x, y h , ∀h ∈ S) space, sequentially treating a single variable yjs for j ∈ Jb and s ∈ S as binary, and the remaining y-variables as continuous. Exploring the dual angular structure of the deterministic equivalent form, Carøe and Tind [13] proposed the generation of cuts for P s in the (x, y s ) space for each scenario s ∈ S. This is valid because all the y s , s ∈ S, are independent, and by the assumption of relatively complete recourse, the projection of P s and the projection of the original solution set on the (x, y s ) space are the same. To generate such cuts for a current solution (¯ x, y¯s ) for which the value for y¯qs , q ∈ Jb , is fractional, consider the disjunction P s = P0s ∪ P1s , where

414


P0s = {Ax ≥ b

← τ0

(12.8a)

Ts x + Ws y s ≥ rs

← λ0

(12.8b)

− yqs ≥ 0}

← ς0

(12.8c)

← τ1

(12.8d)

Ts x + Ws y s ≥ rs

← λ1

(12.8e)

yqs ≥ 1}

← ς1

(12.8f)

and P1s = {Ax ≥ b

with the associated multipliers as shown above, and where we assume that the nonnegativity and the bounding constraints for x and y are absorbed in Ax ≥ b and Ts x + Ws y s ≥ rs , respectively. Define eq as the qth unit row vector. A convexification cutting plane for the disjunction (12.8) is generated by solving the following linear program. Minimize α¯ x + β y¯s − γ subject to: α − τh A − λh Ts ≥ 0,

(12.9a) ∀h = 0, 1

(12.9b)

β + ς0 eq − λ0 Ws ≥ 0

(12.9c)

β − ς1 eq − λ1 Ws ≥ 0

(12.9d)

τ0 b + λ0 rs − γ ≥ 0

(12.9e)

ς1 + τ1 b + λ1 rs − γ ≥ 0

(12.9f)

α¯ x + β y¯s − γ ≥ −1

(12.9g)

τ, λ, ς ≥ 0, α, β, γ unrestricted.

(12.9h)

¯ γ¯ , λ, ¯ τ¯, ς¯) be an optimal solution of (12.9). If the objective value Let (¯ α, β, obtained for (12.9a) is negative (i.e., equals −1 due to (12.9g)), the inequality ¯ s ≥ γ¯ α ¯ x + βy

(12.10)

eliminates the current fractional solution. Carøe and Tind observe that if the recourse matrix and technology matrix are fixed, (12.9b)—(12.9d) will be the same for all scenarios. Hence, α ¯ and β¯ obtained for one scenario will satisfy these constraints for all scenarios, and the cuts generated for one scenario are valid for all scenarios by updating ¯ s0 ≥ γs0 is valid for scenario only the right-hand sides. In particular, α ¯ x + βy ¯ 0 (rs0 − rs ), λ ¯ 1 (rs0 − rs )}. When only the s0 , ∀s0 ∈ S, where γs0 = γ¯ + min{λ


415

recourse matrix is fixed but the technology matrix is stochastic, constraints (12.9c) and (12.9d) are satisfied by β¯ for all s0 ∈ S, and the following linear program can be solved to obtain coefficients α ˜ and γ˜ for deriving a valid inequality for scenario s0 . Minimize α¯ x−γ

(12.11a)

¯ h Ts0 , subject to: α − τh A ≥ λ

∀h = 0, 1

(12.11b)

¯ 0 rs0 τ0 b − γ ≥ −λ

(12.11c)

¯ 1 rs0 τ1 b − γ ≥ −¯ ς1 − λ

(12.11d)

α¯ x − γ ≥ −1

(12.11e)

τ ≥ 0, α, γ unrestricted.

(12.11f)

¯ Comparing this with (12.9b)—(12.9h), because the β-value is already at hand, the terms and constraints related to β and Ws are dropped, and this reduces the size of the problem. Letting (˜ α, γ˜ , τ˜) be an optimal solution to (12.11), ¯ s0 ≥ γ˜ is valid for scenario s0 , s0 ∈ S. the cut αx ˜ + βy In a similar fashion, if T and r are fixed and W is stochastic, or if W and r are fixed and T is stochastic, then a valid inequality for scenario s0 can be obtained by revising the valid inequality (12.10) for scenario s as above. Proposition 12.1. Let (12.10) be a valid inequality obtained for scenario s by solving (12.9). We have: (a) If the technology matrix T and the right-hand side r are fixed, and the recourse matrix W is stochastic, then 0

α ¯ x + βs0 y s ≥ γ¯

(12.12a)

is a valid inequality for scenario s0 , where ¯ 0 (Ws0 − Ws ), λ ¯ 1 (Ws0 − Ws )}, βs0 = β¯ + max{λ

(12.12b)

and where the max-operation for vectors is performed componentwise. (b) Similarly, if the recourse matrix W and the right-hand side r are fixed, and the technology matrix T is stochastic, then ¯ s0 ≥ γ¯ αs0 x + βy

(12.12c)

is a valid inequality for scenario s0 , where ¯ 0 (Ts0 − Ts ), λ ¯ 1 (Ts0 − Ts )}. αs0 = α ¯ + max{λ

(12.12d)

Proof. The proof follows the same line as that of the fixed T and W case in Carøe and Tind [13]. In the case of fixed T and r, we have that the coefficients

416


α ¯ and γ¯ satisfy constraints (12.9b), (12.9e), and (12.9f) for s, where Ts and rs are indistinguishable for all scenarios. Furthermore, from (12.12b), ¯ 0 Ws + λ ¯ 0 (Ws0 − Ws ) = −¯ βs0 ≥ −¯ ς0 eq + λ ς0 eq + λ0 Ws0 ¯ 1 Ws + λ ¯ 1 (Ws0 − Ws ) = ς¯1 eq + λ1 Ws0 , and βs0 ≥ ς¯1 eq + λ which satisfies (12.9c) and (12.9d) for scenario s0 . This proves Part (a), and Part (b) can be proved similarly. t u Note that the cuts (12.12a) and (12.12c) preserve the fixed technology matrix and recourse matrix structure, respectively, in addition to having the same right-hand side. Linear programs similar to (12.11) can be constructed to generate valid inequalities when only one of T , W , and r is deterministic and the other two are stochastic. The above approach is based on solving the deterministic equivalent form of the two-stage SMIP. If a subproblem having the set of variables (x, y s ) is solved for each scenario s sequentially, then the validity of the above cuts still holds true. When using the L-shaped method (or Benders’ decomposition) to solve two-stage stochastic programs, we need the value functions of the Benders subproblems (i.e., the second-stage problems) to be convex. However, if the second-stage problems contain binary or integer variables, their value functions are in general nonconvex and discontinuous. This then requires a relaxation of the binary restrictions in the second-stage problems and an accompanying convexification process. When solving the second-stage problems, a given solution x ¯ is available from the first-stage problem and becomes part of the right-hand sides of the second-stage problems. Valid inequalities obtained in this context will be in the form of β¯s y s ≥ γ¯s . It is helpful to lift these (possibly facetial) cuts to be valid in the (x, y s ) space, in the form of αs x + β¯s y s ≥ γ¯s + αs x ¯. By treating x as a variable, these cuts can be reused in subsequent Benders iterations by updating the first-stage solutions from the Benders master problem, as detailed in Sherali and Fraticelli [39]. This cut lifting is not well posed when x is continuous. If x is restricted to be binary, then the feasible x-solutions are always facial to the convex hull of the two-stage constraint set, and we can lift a valid inequality using the disjunction ⎧ ⎫ ⎧ ⎫ Ax ≥ b Ax ≥ b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬_⎨ ⎬ s s Ts x + Ws y ≥ rs Ts x + Ws y ≥ rs . (12.13) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ −xi ≥ 0 xi ≥ 1 The lifted cut can be obtained via the solution to the following linear program, where β¯s and γ¯s are cut coefficients already obtained in the y s space.


417

Minimize αs x ¯

(12.14a)

subject to: αs + ς0 ei − τ0 A − λ0 Ts ≥ 0

(12.14b)

αs − ς1 ei − τ1 A − λ1 Ts ≥ 0

(12.14c)

− λh Ws ≥ −β¯s ,

(12.14d)

∀h = 0, 1

− αs x ¯ + τ0 b + λ0 rs ≥ γ¯s

(12.14e)

− αs x ¯ + ς1 + τ1 b + λ1 rs ≥ γ¯s

(12.14f)

αs x ¯ ≥ −1

(12.14g)

τ, λ, ς ≥ 0, αs unrestricted.

(12.14h)

Also, in the purely binary first-stage problem setting, if additionally the recourse matrix is fixed, Sen and Higle [35] propose to lift cuts for second-stage problems in the (x, y s )-space, such that the cuts share the same coefficients ¯ s ≥ γs − αs x. β¯ for the y-variables. These cuts are then in the form of βy 3 This property is named common cut coefficients (C ), and is established as follows. Given a first-stage solution x ¯ and a fixed recourse matrix W , consider the disjunction ( ) ( ) W y s ≥ rs − Ts x ¯ _ W y s ≥ rs − Ts x ¯ . (12.15) −yqs ≥ 0 yqs ≥ 1 The common cut coefficients β¯ are obtained via the optimal value for β given by the following linear program, where y¯s , ∀s ∈ S, are the current second-stage solutions. X Minimize ps (β y¯s − πs ) (12.16a) s

subject to: β + ς0 eq − λ0 W ≥ 0

(12.16b)

β − ς1 eq − λ1 W ≥ 0

(12.16c)

λ0 (rs − Ts x ¯) − πs ≥ 0,

∀s ∈ S

ς1 + λ1 (rs − Ts x ¯) − πs ≥ 0, X ps (β y¯s − πs ) ≥ −1,

∀s ∈ S

(12.16d) (12.16e) (12.16f)

s

π, λ, ς ≥ 0, β unrestricted. ¯ be an optimal value for λ in Problem (12.16), the cut Letting λ

(12.16g)

418


¯ 0 Ts x, λ ¯ 1 rs + ς1 − λ ¯ 1 Ts x} ¯ ≥ min{λ ¯ 0 rs − λ βy

(12.17)

is then valid for scenario s, and its piecewise linear concave right-hand side ¯ ≡ {x ∈ can be convexified using its epigraph over the bounded region X n1 R+ |Ax ≥ b}. Denote this epigraph as s ¯ ΠX ¯ ≡ {( , x)|x ∈ X,

≥ Rs (x)},

(12.18)

¯ 0 rs − λ ¯ 0 Ts x, λ ¯ 1 rs + ς1 − λ ¯ 1 Ts x}. where Rs (x) ≡ min{λ ¯ In this process, for x ∈ X, we can find a lower bound for Rs (x), say, L. Each affine function in Rs (x) is then translated by L, if necessary, to ensure that ≥ 0. This will translate the convex hull of the epigraph as well. After convexification cuts are obtained, they can be translated back by −L to recover the original values. So without loss of generality, assume that L = 0. s We can then represent ΠX ¯ as the following disjunction (

¯ 0 Ts x ¯ 0 rs − λ ≥λ Ax ≥ b

)

_

(

¯ 1 rs + ς¯1 − λ ¯ 1 Ts x ≥λ Ax ≥ b

)

.

(12.19)

The right-hand side of (12.17) can then be convexified based on this disjunction by solving the following linear program for each scenario s. Minimize σs x ¯ + υs − δs

(12.20a)

¯ h Ts ) ≥ 0, subject to: σs − τh A − θh (λ υs − θh ≥ 0,

∀h = 0, 1

(12.20b)

∀h = 0, 1

(12.20c)

¯ 0 rs ) − δs ≥ 0 τ0 b + θ0 (λ

(12.20d)

¯ 1 rs ) + θ1 ς¯1 − δs ≥ 0 τ1 b + θ1 (λ

(12.20e)

θ0 + θ1 = 1

(12.20f)

θ, τ ≥ 0 σ, δ, υ unrestricted.

(12.20g)

Note that from (12.20c) and (12.20f), we have that υs > 0. For an optimal s extreme point solution (¯ σs , υ¯s , δ¯s ), we then have clconv (ΠX ¯ ) = {( , x)|x ∈ ¯ X, ≥ (δ¯s /¯ υs ) − (¯ σs /¯ υs )x}. This completes the convexification of the piecewise linear concave Rs (x). Letting α ¯s = σ ¯s /¯ υs and γ¯s = δ¯s /¯ υs , the inequality ¯ ≥ γ¯s − α ¯sx βy

(12.21)

is then valid for scenario s, ∀s ∈ S, and all scenarios share the same coefficient β¯ for y in the new cuts, which need to be stored only once. The disjunctive decomposition (D2 ) method by Sen and Higle [35] thus applies Benders’ decomposition to solve two-stage stochastic programs, using


419

the above C 3 -scheme embodied by (12.16) and (12.20) to solve the secondstage subproblems. The two approaches by Carøe and Tind [13] and Sen and Higle [35] directly stem from disjunctive programming. Using another approach, the reformulation-linearization technique (RLT) (see Sherali and Adams [37, 38]), we can sequentially construct (partial) convex hulls for the second-stage subproblems. Sherali and Fraticelli [39] proposed a method for solving two-stage problems having purely binary first-stage variables using a modified Benders’decomposition scheme, where the subproblems are solved by adding RLT cuts generated in the (x, y s ) space. The idea is to store the cuts as functions of x, so that at each Benders iteration, when new x-values are obtained from the master (first-stage) problem, the cutting planes obtained previously from the second-stage subproblem solutions can be reused in the subsequent subproblems of the same scenario simply by updating the x-values. In a Benders’-decomposition setting, each second-stage problem is solved independently; hence, we omit the superscript s for y-variables when no confusion arises. Furthermore, denote the qth column of matrix Ws as Wsq , and ˜ s . Correspondingly, the matrix formed by the remaining columns of Ws as W denote variable y without the qth element as y˜ and let β˜ be its associated coefficient; that is, βy ≡ β˜y˜ + βq yq . To derive cuts in the (x, y) space, that is, to introduce x as a variable into the cut generation process, we include the bounding constraint 0 ≤ x ≤ e into the second stage. By the RLT process, given a solution (¯ x, y¯), which has y¯q fractional for some q ∈ Jb , we multiply 0 ≤ x ≤ e and the constraints in the second stage by the factors yq and (1 − yq ) to obtain a system in a higher-dimensional space (x, y, z x , z y ), including the new RLT variables z x and z y to represent the resulting nonlinear products; that is, z x ≡ xyq and z y ≡ y˜yq . (12.22) Denote the resulting system as ˜ s z y ≥ (rs − Wsq )yq Ts z x + W

← φ0

(12.23a)

˜ s z y ≥ rs − Ts x − W ˜ s y˜ − rs yq − Ts z x − W

← φ1

(12.23b)

Γz z x ≥ hb − Γx x − Γy yq ,

← φb .

(12.23c)

The constraints (12.23a) and (12.23b) are obtained by multiplying (12.1g) (for scenario s) by yq and (1 − yq ), respectively. The constraints (12.23c) are obtained by multiplying the bounding constraints of 0 ≤ x ≤ e by yq and (1 − yq ), where Γz , Γx , and Γy are used to denote the resulting coefficient matrices for z x , x, and yq , respectively. Associating the dual multipliers φ as indicated in (12.23), the solution (¯ αs , β¯s , γ¯s ) of the following linear program yields the valid inequality β¯s y ≥ γ¯s − α ¯ s x for scenario s in the projected space of the original variables (x, y).

420


Minimize β˜s y˜¯ + βqs y¯q + αs x ¯ − γs subject to: (φ0 − φ1 )Ts + φb Γz = 0 ˜s = 0 (φ0 − φ1 )W αs = φ1 Ts + φb Γx ˜s β˜s = φ1 W φ0 (Wsq

βqs = − rs ) + φ1 rs + φb Γy γs = φ1 rs + φb hb ¯ − γs ≥ −1 β˜s y˜¯ + βqs y¯q + αs x φ ≥ 0, αs , βs , γs unrestricted.

(12.24a) (12.24b) (12.24c) (12.24d) (12.24e) (12.24f) (12.24g) (12.24h) (12.24i)

Proposition 12.2. If the technology matrix T and recourse matrix W are ˜s y˜ + β¯qs yq ≥ γ¯s − α fixed, and β¯ ¯ s x is a valid inequality for scenario s obtained from (12.24), then ¯sx (12.25) β¯˜s y˜ + βqs0 yq ≥ γs0 − α is valid for scenario s0 , where βqs0 = β¯qs + φ¯0 (rs − rs0 ) and γs0 = γ¯s + φ¯1 (rs0 − rs ).

Proof. Follows from the feasibility of (12.24b)—(12.24i).

t u

Note that the cut (12.25) is valid for scenario s0 , but it disturbs the fixed recourse structure. Hence, if applied, it should not be included in the cut generation process in later iterations, which should continue to use the fixed W . However, ultimately, to obtain a convexification of the second-stage problem, such cuts would need to be included in the fashion discussed by Sherali and Fraticelli [39]. The following proposition provides a way to derive valid inequalities and retain a fixed recourse, in a more general setting of a stochastic technology matrix. Proposition 12.3. Let the recourse matrix W be fixed, and the technology ¯ as an matrix Ts and right-hand side rs be stochastic. Denote (¯ αs , β¯s , γ¯s , φ) optimal solution obtained for Problem (12.24). Solve the following linear program corresponding to another scenario s0 6= s. Minimize 0 subject to: φTs0 = φ¯1 Ts0 − φ¯b Γz ˜ ˜ = φ¯1 W φW q φ(W − rs0 ) = φ¯0 (W q − rs0 ) φ ≥ 0.

(12.26a) (12.26b) (12.26c) (12.26d) (12.26e)

If (12.26) is feasible, then β¯s y ≥ γs0 − αs0 x

(12.27a)


421

is a valid inequality for scenario s0 , where αs0 = φ¯1 Ts0 + φ¯b Γx

and

γs0 = φ¯1 rs0 + φ¯b hb .

(12.27b)

Proof. The proof again follows from the feasibility to (12.24b)—(12.24i) for scenario s0 by revising φ0 to φ and retaining φ1 = φ¯1 and φb = φ¯b , where (12.24e) remains the same, and (12.24d) and (12.24g) are satisfied by (12.27b). Furthermore, (12.24b), (12.24c), and (12.24f), are satisfied by (12.26b), (12.26c), and (12.26d), respectively, and the normalization constraint (12.24h) is inconsequential. t u The purpose of constraint (12.26d) is to retain the same βq value for all new cuts generated, so as to keep the fixed recourse structure if the cuts are to be inherited. It can be replaced by other normalization constraints, and a valid cut can be generated using the correspondingly computed (αs0 , βs0 , γs0 ) values. However, we may no longer have βs0 = β in the resulting cut, thereby losing the fixed recourse structure. In a Benders’-decomposition context where subproblems are solved using a cutting plane method, and where these cutting planes are valid in the (x, y)space for conv {(x, y) | Ts x+W y ≥ rs , 0 ≤ x ≤ e, y ≥ 0, yj ∈ {0, 1}, ∀j ∈ Jb }, we can append these cuts along with any possibly added bounding constraints for the y-variables to Ts x + W y ≥ rs , and Sherali and Fraticelli [39] have shown that Benders cuts generated using the dual solution of the augmented LP relaxation system are valid optimality cuts in terms of the first-stage variables. Using this idea and disjunctive programming, Sen and Sherali [36] develop a disjunctive decomposition-based branch-and-cut (D2 -BAC) approach, where subproblems are partially solved using branch-and-cut, and where optimality cuts applied in the first-stage master problem are generated using the disjunctive programming concept. In the B&B tree for any subproblem in this process, each node is an LP relaxation of the subproblem that includes some node-dependent bounding constraints for y. Assuming that all the nodes are associated with feasible LP relaxations and are fathomed only by the bound computations, then at least one terminal node corresponds to an optimal solution. Let Qs denote the set of terminal nodes of the tree that have been explored for the subproblem for scenario s, and let zqls and zqus denote the lower and upper bounds for y in the nodal subproblem q, ¯ qs , ψ¯qls , and ψ¯qus be dual solutions associated with the for q ∈ Qs . Let λ constraint set W y ≥ rs − Ts x and the lower and upper bounding constraints for y, respectively, in the nodal subproblem for node q. We then obtain the following disjunction: ¯ qs [rs − Ts x] + ψ¯qls zqls − ψ¯qus zqus η≥λ

for at least one q ∈ Qs . (12.28)

Similar to the convexification process for (12.17), we can use the following disjunction to generate a disjunctive cut for (12.28),

422


¾ _ ½η + λ ¯ qs Ts x ≥ λ ¯ qs rs + ψ¯qls zqls − ψ¯qus zqus . Ax ≥ b

(12.29)

q∈Qs

Denote η¯ as a current estimate lower bound for the second-stage value function, possibly obtained from previous iterations, and let (¯ σs , υ¯s , δ¯s ) be an optimal extreme point solution of the following linear program. Minimize σs x ¯ + υs η¯ − δs ¯ qs Ts ≥ 0, ∀q ∈ Qs subject to: σs − τq A − θq λ υs − θq ≥ 0, ∀q ∈ Qs ¯ qs rs + ψ¯qls zqls − ψ¯qus zqus ) − δs ≥ 0, τq b + θq (λ X

(12.30a) (12.30b) (12.30c) ∀q ∈ Qs (12.30d)

θq = 1

(12.30e)

q∈Qs

θ, τ ≥ 0 σ, δ, υ unrestricted.

(12.30f)

Again, due to (12.30c) and (12.30e), we have that υ¯s > 0. A disjunctive cut that provides a lower bound on the second-stage value function can then be obtained in the form of ηs ≥ γ¯s − α ¯ s x, where α ¯s = σ ¯s /¯ υs and γ¯s = δ¯s /¯ υs .

12.3.2 Pure Continuous and Mixed 0—1 First-Stage Problems Following the cutting plane game concept of Jeroslow [18], the disjunctive and RLT cut generation processes finitely solve the second-stage subproblems. Therefore, finite convergence of the D2 algorithm of Sen and Higle [35], and of the modified Benders’-decomposition algorithm of Sherali and Fraticelli [39], is afforded by the finite number of feasible first-stage solutions and by the finite cutting plane generation processes for solving subproblems. If the first stage contains continuous variables, however, a feasible first-stage solution x ¯ may not be facial with respect to its bounding constraints, and the algorithms of Section 12.3.1 would no longer assure convergence. To retain the facial property of the first-stage solutions, Ntaimo and Sen [27] and Sherali and Zhu [41] propose to build a B&B tree via a partitioning process in the projected space of the bounded first-stage variables so as to ultimately induce the aforementioned facial property. Using the D2 algorithm, when x ¯ is at a vertex of its bounding region, for the right-hand side of the cut (12.21) generated by (12.16) and (12.20), we will have ¯ 0 rs − λ ¯ 0 Ts x, λ ¯ 1 rs + ς1 − λ ¯ 1 Ts x}(≡ Rs (x)); γ¯s − α ¯ s x = min{λ

(12.31)


423

otherwise, this equality may be violated. Ntaimo and Sen [27] then propose a finitely convergent D2 -based branch-and-cut (D2 -CBAC) algorithm for problems containing purely continuous first-stage variables. This algorithm constructs a B&B tree defined on the first-stage feasible region, applies the D2 algorithm at each node, selects some scenario s and iteration k such that (12.31) is mostly violated, and partitions based on the disjunction prompted ¯ 0 rs − λ ¯ 0 Ts x ≥ λ ¯ 1 rs + ς1 − λ ¯ 1 Ts x on one child node by (12.31), enforcing λ ¯ 0 rs − λ ¯ 0 Ts x ≤ λ ¯ 1 rs + ς1 − λ ¯ 1 Ts x on the other. The convexification and λ ¯ ≥λ ¯ k rk − λ ¯k T k x cut (12.21) at the parent node is accordingly updated to βy 0 s 0 s k k k k ¯ ≥λ ¯ rs + ς − λ ¯ T x, at the two child nodes, respectively. and βy 1 1 1 s Because there are finitely many disjunctive variables in the second stage, for some iteration of the embedded D2 algorithm, there are a finite number of right-hand sides of (12.17) that can be constructed; hence, there are finitely many partitions of the first-stage feasible region to consider, which leads to finite convergence of the D2 -CBAC algorithm. If the first stage further contains both continuous and binary variables, Sherali and Zhu [41] propose a decomposition-based B&B (DBAB) algorithm that is guaranteed to converge to an optimal solution. They assume relative complete recourse with respect to some bounding hyperrectangle of x. The branch-and-bound tree is again defined on the projected space of the bounded first-stage variables, where lower bounds for the nodal problems are computed by applying a modified Benders’ method extended from Sherali and Fraticelli [39], but defined on a subdivision of the original bounding hyperrectangle for x, and the Benders subproblems are derived based on partial convex hull representations in the (x, y s )-spaces using the second-stage constraints and the current bounding constraints for x. For some given feasible first-stage solution x ¯, because x = x ¯ may not be facial with respect to its bounding constraints, the Benders subproblems are shown to define lower bounds for second-stage value functions. Therefore, any resulting Benders master problem provides a lower bound for the original stochastic program defined over the same hyperrectangle, and yields the same objective value if x ¯ is a vertex of the defining region. In the branch-and-bound process of the DBAB algorithm, a node yielding the least lower bound is selected for branching at each partitioning step. Hence, the nodal objective value provides a lower bound for the original twostage problem. In the partitioning step, the first-stage continuous and binary variables are dealt with differently. A variable xp whose current value is most in-between its bounds is selected as the partitioning variable. If p ∈ Ib , then xp is fixed at 0 and 1 in the two child nodes, respectively; otherwise, xp is a continuous variable, and its current value x ¯p (or the midpoint of the current bounding interval for xp ) is used as the lower and upper bounds for the two child nodes, respectively. Therefore, barring finite convergence, along any infinite branch of the branch-and-bound tree, there will exist a subsequence of the selected nodes such that the bounding interval for some continuous variable xp is partitioned infinitely many times and such that the limiting

424


value x ¯p coincides with one of its bounds. As xp was selected whenever its value was most in-between its bounds, in the limit, all x ¯i , ∀i = 1, . . . , n1 would coincide with one of their bounds, and hence, x ¯ would be a vertex of the limiting bounding hyperrectangle, thereby providing an upper bounding solution for the original two-stage stochastic program. Together with the node selection rule, the partitioning process therefore guarantees convergence of the algorithm to an optimal solution. A difficulty in directly implementing the modified Benders’ method by Sherali and Fraticelli [39] at the nodal problems arises due to the fact that when x ¯ is not extremal with respect to its bounds, the solution y¯ obtained for a Benders subproblem defined on a partial convex hull representation in the (x, y)-space may not satisfy its binary restrictions. We thus need to be able to detect whether a Benders subproblem is already solved by such a fractional y¯. This can be achieved as follows. If y¯j ∈ {0, 1}, ∀j ∈ Jb , or if (¯ x, y¯) can be represented as a convex combination of some extreme points of the current partial convex hull defining the Benders subproblem such that these extreme points have binary yj -variables for all j ∈ Jb , then y¯ solves the Benders subproblem. Sherali and Zhu [41] describe a procedure to check this situation in their overall algorithmic scheme. The RLT cuts generated for any given subhyperrectangle are reusable by updating the x-values in subsequent Benders iterations at the same node, and are also inheritable by the subproblems of the child nodes. Likewise, the Benders cuts derived for a given subhyperrectangle can also be inherited by the lower bounding master programs solved for its child nodes.

12.3.3 Connections Between Disjunctive Cuts and RLT Cuts In this section, we demonstrate the connections between the two types of convexification cuts, namely, the disjunctive cuts generated by Problems (12.16) and (12.20) (or the counterpart of problem (12.9) in a stochastic setting), and the RLT cuts generated by Problem (12.24) under fixed recourse. We discuss these cuts in the context of fixed recourse stochastic programs having continuous first-stage variables, and the analysis is similar for the case of discrete first-stage variables. Let x be continuous and bounded by l ≤ x ≤ u at some node of the B&B process performed on the projected x-space as described in Section 12.3.2. Upon multiplying the constraints ˜ y˜ + W q yq ≥ rs Ts x + W

and

l≤x≤u

(12.32)

by yq and (1−yq ), linearizing upon substituting the resulting nonlinear terms using (12.22), the higher-dimensional system (12.23) obtained is as follows.


˜ z y ≥ (rs − W q )yq Ts z x + W ˜ z y ≥ rs − Ts x − W ˜ y˜ − rs yq − Ts z x − W

z x ≥ lyq − z x ≥ −uyq − z x ≥ −lyq + l − x z x ≥ uyq − u + x

425

← φ0

← φ1 ← φxl0 ← φxu0 ← φxl1 ← φxu1 .

(12.33a) (12.33b) (12.33c) (12.33d) (12.33e) (12.33f)

The cut generation problem as a counterpart to (12.24) then takes the following form. Minimize β˜s y¯˜ + βqs y¯q + αs x ¯ − γs subject to: (φ0 − φ1 )Ts + φxl0 − φxu0 − φxl1 + φxu1 = 0 ˜ =0 (φ0 − φ1 )W αs = φ1 Ts + φxl1 − φxu1 ˜ β˜s = φ1 W q

βqs = γs + φ0 (W − rs ) − φxl0 l + φxu0 u γs = φ1 rs + φxl1 l − φxu1 u ¯ − γs ≥ −1 β˜s y˜¯ + βqs y¯q + αs x φ ≥ 0, αs , β˜s , βqs , γs unrestricted,

(12.34a) (12.34b) (12.34c) (12.34d) (12.34e) (12.34f) (12.34g) (12.34h) (12.34i)

where (¯ x, y¯) is the current solution having y¯q , q ∈ Jb , fractional. Note that the resulting cut is of the form α ¯ s x + β¯˜s y˜ + β¯qs yq ≥ γ¯s ,

(12.35)

˜s , β¯qs , γ¯s , φ) ¯ solves Problem (12.34). where (¯ αs , β¯ Proposition 12.4. Problem (12.34) is equivalent to Minimize β˜s y¯˜ + βqs y¯q + αs x ¯ − γs subject to: αs ≥ φ0 Ts + φxl0 − φxu0 αs ≥ φ1 Ts + φxl1 − φxu1 ˜ β˜s ≥ φ0 W

˜ β˜s ≥ φ1 W βqs ≥ γs + φ0 (W q − rs ) − φxl0 l + φxu0 u γs ≤ φ1 rs + φxl1 l − φxu1 u β˜s y˜¯ + βqs y¯q + αs x ¯ − γs ≥ −1 φ ≥ 0, αs , β˜s , βqs , γs unrestricted.

(12.36a) (12.36b) (12.36c) (12.36d) (12.36e) (12.36f) (12.36g) (12.36h) (12.36i)

426


Proof. Noting the objective coefficients of βqs and γs in Problem (12.34), we can equivalently write the equalities of (12.34f) and (12.34g) as the inequalities (12.36f) and (12.36g), respectively, as these constraints will always hold as equalities in any optimal solution. Substituting for the terms involving φ1 in (12.34d) and (12.34e) into (12.34b) and (12.34c), respectively, yields ˜ , that is, (12.36b) and (12.36d) in αs = φ0 Ts + φxl0 − φxu0 and β˜s = φ0 W equality forms. To complete the proof, we now show that at an optimal solution to (12.36), constraints (12.36b)—(12.36e) will hold as equalities as in (12.34b)—(12.34e). Suppose that, on the contrary, we have an optimal solution ˆ˜ βˆ , γˆ , φˆ , φˆ , φˆ ), (ˆ α, β, qs xl1 xl0 − where φ− represent the vector φ except for the elements φxl1 and φˆxl0 , such that α ˆ si = φˆ0 Tsi + φˆxl0i − φˆxu0i > φˆ1 Tsi + φˆxl1i − φˆxu1i , or α ˆ si = φˆ1 Tsi + φˆxl1i − φˆxu1i > φˆ0 Tsi + φˆxl0i − φˆxu0i

(12.37a) (12.37b)

for some i ∈ I, where Tsi is the ith column of Ts . If (12.37a) occurs, then let φ i = [φˆ0 Tsi + φˆxl0i − φˆxu0i ] − [φˆ1 Tsi + φˆxl1i − ˆ˜ β 0 , γ 0 , φ0 , φˆ , φˆ ) such that φˆxu1i ] > 0. Obtain a new solution (ˆ α, β, xl0 − qs

φ0xl1i = φˆxl1i + φ i ,

φ0xl1j = φˆxl1j , γs0

s

xl1

∀j 6= i,

= γˆs + li φ i ,

and

0 βqs = βˆqs + li φ i .

This new solution is feasible, satisfies α ˆ si = φˆ0 Tsi + φˆxl0i − φˆxu0i = φˆ1 Tsi + 0 φxl1i − φˆxu1i , and reduces the objective value by φ i li (1 − y¯q ) > 0. Hence, if ˆ˜ βˆ , γˆ , φˆ , φˆ , φˆ ) cannot be an optimal solution. On (12.37a) holds, (α ˆ , β, qs xl1 xl0 − the other hand, suppose that (12.37b) occurs. Then let φ i = [φˆ1 Tsi + φˆxl1i − φˆxu1i ] − [φˆ0 Tsi + φˆxl0i − φˆxu0i ] > 0. Similarly, we can obtain a new solution ˆ ˜ β 0 , γˆs , φˆxl1 , φ0 , φˆ− ) such that (ˆ α, β, qs xl0 0 φ0xl0i = φˆxl0i + φ i , φ0xl0j = φˆxl0j , ∀j 6= i, and βqs = βˆqs − li φ i .

Again, this new solution is feasible, satisfies α ˆ si = φˆ1 Tsi + φˆxl1i − φˆxu1i = 0 φˆ0 Tsi + φxl0i − φˆxu0i , and reduces the objective value by φ i li y¯q > 0. Hence, ˆ˜ βˆ , γˆ , φˆ , φˆ ) cannot be an optimal solution, if (12.37b) holds, then (ˆ α, β, qs xl1 − either. Therefore, we always have (12.36b) and (12.36c) tight at an optimal solution. Similarly, (12.36d) and (12.36e) always hold as equalities in an optimal solution. t u


427

Now, consider the disjunctive cut generation where Ax ≥ b contains only the bounding constraints l ≤ x ≤ u for x. Applying yqs ∈ {0, 1} in the system (12.32), we obtain the following disjunction for scenario s. ⎧ ⎫ ⎧ ⎫ ˜ y˜s ≥ rs ⎪ ˜ y˜s ≥ rs − W q ⎪ ← φ0 φ1 → ⎪ Ts x + W Ts x + W ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬_⎨ ⎬ φxl1 → x ≥ l ← φxl0 x≥l . φxu1 → ⎪ ← φxu0 −x ≥ −u −x ≥ −u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ s ⎭ λ1 → −yqs ≥ 0 ← λ0 yq ≥ 1 (12.38) Using the associated multipliers, a disjunctive cut can be generated by solving the following problem. Minimize β˜s y¯˜ + βqs y¯q + αs x ¯ − γs subject to: constraints (12.36b)—(12.36e), (12.36h), and βqs ≥ −λ1 βqs ≥ λ0 γs ≤ φ0 (rs − W q ) + φxl0 l − φxu0 u + λ0 γs ≤ φ1 rs + φxl1 l − φxu1 u φ, λ ≥ 0, αs , β˜s , βqs , γs unrestricted.

(12.39a) (12.39b) (12.39c) (12.39d) (12.39e) (12.39f) (12.39g)

Because βqs appears only in constraints (12.39c) and (12.39d), we will have βqs ≡ λ0 ≥ −λ1 in an optimal solution. Constraints (12.39c)—(12.39e) then directly reduce to (12.36f), and Problem (12.39) is exactly the same as Problem (12.36). Therefore, the cut (12.35) generated using a RLT process is indeed also a disjunctive cut. Applying the C 3 result from Sen and Higle [35], we can then generate disjunctive cuts that are valid for the disjunctions (12.38) for all scenarios, so that the cuts contain a common coefficient β for y. Similar to (12.16), the cut coefficients can be obtained by collecting the coefficient matrices of all scenarios and solving the following problem. X Minimize ps (β˜y¯˜s + βq y¯qs + αs x ¯ − γs ) (12.40a) s

subject to: αs ≥ φh Ts + φxlhs − φxuhs , ˜ , ∀h = 0, 1 β˜ ≥ φh W

∀h = 0, 1, ∀s ∈ S

q

βq ≥ γs + φ0 (W − rs ) − φxl0s l + φxu0s u, γs ≤ φ1 rs + φxl1s l − φxu1s u, ∀s ∈ S X ps (β˜y¯˜s + βq y¯qs + αs x ¯ − γs ) ≥ −1

∀s ∈ S

(12.40b)

(12.40c) (12.40d) (12.40e) (12.40f)

s

˜ βq , γs unrestricted. φ ≥ 0, αs , β,

(12.40g)

428


¯˜ β¯ , α For an optimal solution (β, ¯s , ∀s), β¯˜y˜s + β¯q yqs ≥ γ¯s − α ¯ s x, ∀s ∈ S q ¯s, γ are then valid inequalities for disjunctions (12.38) for all scenarios s ∈ S, and ¯˜ β¯ ) for y. they share the same coefficient β¯ ≡ (β, q We close this section by emphasizing that the efficacy of cutting plane methods depends on problem structures. They are most efficient when tight representations are available using relatively few cuts. Ntaimo and Sen [26] have successfully applied the disjunctive decomposition (D2 ) method ([35]) on stochastic server location problems having as many as 1,000,010 binary variables and 120,010 constraints, using a Sun 280R with UltraSPARC-III+ cpus running at 900 MHz. For most of the nontrivial problems, the benchmarking commercial MIP software package CPLEX 7.0 failed to solve the problems. Another reason for this method to be so effective is due to the cut coefficient sharing property. As we have mentioned, in practice, cut coefficient sharing and reuse are important in saving computation, and greatly enhance the efficacy of cutting plane methods.

12.4 Structural Enumeration Using a Fixed Technology Matrix In this section, we assume the following. A1. The first-stage feasible region X is bounded, and thus compact. A2. For any x ∈ X, the second-stage problems are feasible (relative complete recourse). A3. For any x ∈ X, the second-stage value functions are bounded.

A4. The second-stage variables are purely integer; that is, y ∈ Zn+2 . A5. The technology matrix T is fixed (i.e., deterministic). A6. All elements in the recourse matrices Ws are integral. (Rational elements can be scaled to obtain integral elements.) Assumption A1 assures finite termination for the enumeration schemes used in the algorithms in this section. Assumptions A2 and A3 imply that 2 fs (x) and u(rs − T x), ∀u ∈ Rm + , are finite. Using Assumption A5, we can transform the second-stage value functions as defined on the space of the so-called “tender variables,” −T x, which we denote as χ. That is, Fs (χ) = min{gs y | Ws y ≥ rs + χ, y ∈ Zn+2 } = fs (x).

(12.41)

From Assumptions A4 and A6, we have that Ws y ≥ t implies Ws y ≥ dte, where d·e and b·c, respectively, denote the componentwise rounded-up and rounded-down forms of a vector. Therefore, we have that fs (x) is constant on the sets


429

{x ∈ Rn+1 : drs − T xe = κ} = {x : κ − rs − e < −T x ≤ κ − rs },

∀κ ∈ Zm2 , (12.42a)

and Fs (χ) is constant on the hyperrectangles m2 Y

(κj − rsj − 1, κj − rsj ],

j=1

∀κ ∈ Zm2 .

(12.42b)

T |S|m2 Therefore, for some integral , P vector k ≡ (k11 , . . . , ksj , . . . , k|S|m2 ) ∈ Z the expectation Q(x) ≡ s ps fs (x) is constant over the intersection of the sets (12.42a):

C(k) ≡

m2 \ \

{x : ksj − rsj − 1 < −Tj x ≤ ksj − rsj },

s∈S j=1

∀k ∈ Z|S|m2 (12.43a)

P and Q(χ) ≡ s ps Fs (χ) is constant over the intersection of the hyperrectangles (12.42b): C(k) ≡

m2 \Y

(ksj − rsj − 1, ksj − rsj ],

s∈S j=1

∀k ∈ Z|S|m2 .

(12.43b)

Based on the above observation that function values are constant over regions, Schultz et al. [32] and Ahmed et al. [1] developed algorithms that construct partitions of the first-stage feasible region using C(·) and C(·), respectively, and evaluate expected values, Q(·) and Q(·), on these partitions. The algorithm in the former paper operates over the x-space, whereas that in the latter works in the χ-space. Under Assumption A1 that the first-stage feasible region X is compact, Schultz et al. [32] show that there are a finite number of vertices of the sets C(·) ∩ X, and these vertices contain an optimal solution for x. (If X is not compact, this vertex set is only countable, and a level set is used to bound the feasible region to guarantee finite termination.) Aside from A1—A6, Schultz et al. [32] further assume the following. A7. The recourse matrix W is fixed. A8. The first-stage variables are continuous; that is, Ib = Iint = ∅. The purpose of these assumptions is to use the Gr¨ obner-basis method from computational algebra to evaluate the second-stage value functions. Although the computation of Gröbner bases is expensive, it only uses the T and W matrices, and does not rely on the right-hand-side values. Hence, for fixed T and W , Gröbner bases need to be computed only once, and then for each different right-hand-side value, the second-stage function evaluation, which reduces to a single generalized division, is very easy. Using this idea, the objective value cx + Q(x) is evaluated at each candidate point from the finite vertex set using Gröbner bases, and after enumerating all these candidate

430


points, the one that yields the lowest function value is an optimal solution to the problem. Improved versions of this algorithm are proposed to reduce the number of candidate points that need to be evaluated. The explicit enumeration approach is still quite expensive for nontrivial problems. Using a systematic enumeration scheme such as B&B to evaluate the partitions is more practical. However, in the x-space, the discontinuities between these partitions are not orthogonal to the (x-) variable axes, thus branching on x-variables would result in discontinuities in the interior of the hyperrectangles defining the nodal problems. To circumvent this difficulty, Ahmed et al. [1] propose to transform the problem into the space of the tender variable vector χ: X TP: min{cx + ps Fs (χ) | x ∈ X, T x = χ, and (12.41)}, (12.44) s

so that the partitioned hyperrectangles are of the form (12.43b), having constant second-stage function values, and the discontinuities between these hyperrectangles are orthogonal to the (χ-) variable axes. They show that if χ∗ is an optimal solution to (12.44), then x∗ ∈ argmin{cx | x ∈ X, T x = χ∗ } is an optimal solution of (12.1). Their B&B algorithm can handle general first-stage variables and scenariodependent recourse. Hence, Assumptions Qm2A7 kandk A8 are not needed. For each hyperrectangular partition P k ≡ j=1 (lj , uj ] in the form of (12.42b), because Fs (χ) is lower semicontinuous and nondecreasing, we can obtain a lower bound on the two-stage problem defined on P k by solving the following problem. Minimize cx + η subject to x ∈ X, T x = χ

lk ≤ χ ≤ uk X ps Fs (lk + ), η≥

(12.45a) (12.45b) (12.45c) (12.45d)

s∈S

where Fs (·) is as defined in (12.41), and is calculated a priori as a sufficiently small number such that Fs (·) is constant over (lk , lk + ]. The value for is decided as follows. Along each axis j, ∀j = 1, . . . , m2 , within the unit interval (k1j − r1j − 1, k1j − r1j ] for some k1j ∈ Z, the candidate point of discontinuity for each scenario s is identified as bk1j − r1j + rsj c − rsj , ∀s ∈ S. These points of discontinuity repeat the same pattern to the right of k1j − r1j with a unit period. It is then sufficient to sort these points, and obtain the smallest interval as j along axis j. The final -value is chosen as strictly smaller than each j . At the node selection step, the partition that yields the least lower bound is selected for further branching. At the branching step, the branching variable χj 0 is selected such that χj 0 + rsj 0 is an integer greater than lj 0 and Q(χ) is


431

discontinuous at χj 0 for some scenario s. Let y¯s be the solution obtained from (12.45) for scenario s. For each j = 1, . . . , m2 , let the pj ≡ mins∈S {(Ws y¯s )j − rsj }, and let the variable χj 0 be chosen such that pj 0 lies most in-between its bounds ljk0 and ukj0 . Partition P k is then branched along axis j 0 at the value pj 0 . Using such a B&B scheme, the algorithm therefore avoids an exhaustive enumeration of the partitions, yet guarantees finite convergence. This algorithm is applied by Ahmed and Garcia [2] to a dynamic acquisition and assignment problem. Problems up to the size of 24,027 variables (including 24,009 binary variables), 10,518 constraints, and 46,545 nonzeros were randomly generated to test the efficacy of the algorithm. On a Sun Sparc Ultra60 workstation, 85% of the problems were solved within the specified tolerance. The remaining prematurely terminated problems reached an average gap of 2.05% for those due to time limitation and 1.10% for those due to memory or node limitation. Using CPLEX 7.0, however, only 27% of the problems were solved. Similar to Schultz et al. [32], Kong et al. [22] examine integer programs that differ only in their right-hand sides. In lieu of Gr¨ obner bases, their algorithms use stored value functions of parameterized integer programs. In addition to Assumptions A1—A7, they further assume the following. A9. The second-stage objective coefficient vector g is fixed. A10. The first-stage variables are purely integer-restricted; that is, x ∈ Zn+1 . A11. All elements in A, T , b, and rs , ∀s ∈ S, are integral. The first- and second-stage problems now both belong to the type of pure integer programs that have integral coefficient matrices and right-hand-side , this type of integer vectors. In general, given a coefficient matrix G ∈ Zm×n + program can be expressed as (V F ) : z(ζ) = min{dx | Gx ≥ ζ, x ∈ Zn+ },

for ζ ∈ Zm ,

(12.46)

where z(·) : Zm 7→ Z is its value function. z(·) is nondecreasing over Zm and subadditive over D = {ζ ∈ Zm | {x ∈ Zn+ | Gx ≥ ζ} 6= ∅}. That is, for ζ1 , ζ2 ∈ D, ζ1 + ζ2 ∈ D implies that z(ζ1 ) + z(ζ2 ) ≥ z(ζ1 + ζ2 ). (The authors use the superadditivity property for maximization problems. We chose to use subadditivity for minimization problems to maintain consistency in exposition.) Using these properties, Kong et al. [22] develop an integer programmingbased algorithm and a dynamic programming-based algorithm that compute the value of z(ζ) for each ζ in a finite set Υ . In the integer programming-based algorithm, at each iteration k, some ζ k ∈ Υ is selected, and the corresponding integer program min{dx | Gx ≥ ζ k , x ∈ Zn+ } is solved to obtain a solution x ˆk . Based on the value of x ˆk , other ζ ∈ Υ are then selected to have their lower bounds, l(ζ), and upper bounds, u(ζ), updated using the nondecreasing and subadditivity properties of z(ζ), until l(ζ) = u(ζ), for all ζ ∈ Υ , at which point, z(ζ) = l(ζ) = u(ζ) are available for all ζ ∈ Υ . The dynamic

432


programming-based approach does not solve any integer program, but is only applicable when G is nonpositive. Denote Υj to be the set of ζ ∈ Υ such S that ζ ≤ Gj . This algorithm uses the initial condition z(ζ) = 0, ∀ζ ∈ Υ \ nj=1 Υj , and the Snrecursive function z(ζ) = min{dj + z(ζ + Gj ) : Gj ∈ Υ, j = 1, . . . , n}, ∀ζ ∈ j=1 Υj . Naturally, both algorithms work better when the size of Υ is small. To use these value function evaluation algorithms to solve two-stage SMIPs, denote the first-stage value function as z1 (ζ1 ) = min{cx | Ax ≥ b, T x ≥ ζ1 , x ∈ Zn+1 }, ∀ζ1 ∈ Υ 1 ≡ {ζ1 ∈ Zm2 | ζ1 = T x for some x ∈ X}, and denote the second-stage value function as z2 (ζ2 ) = min{gy | W y ≥ ζ2 , y ∈ Zn2 + }, ∀ζ2 ∈ Υ 2 ≡

[ [

ζ1 ∈Υ 1 s∈S

{rs − ζ1 }.

These function values for the first- and second-stage problems are stored for all possible right-hand-side values in Υ 1 and Υ 2 , respectively. (Due to the storage requirement, this approach is more suitable for problems having a relatively small number of rows.) After storing these value function responses, the feasible region for χ is systematically searched to obtain an optimal value χ∗ . A finitely convergent B&B search scheme is proposed. At node k defined on a hyperrectangle [lk , uk ], a lower bound LB k and an upper bound U B k are computed as X LB k = z1 (lk ) + ps z2 (rs − uk ), s

and

U B k = z1 (uk ) +

X s

ps z2 (rs − lk ).

Branching is performed by partitioning the current hyperrectangle by bisecting some selected axis, which ensures convergence.

12.5 Conclusion In this chapter, we have reviewed some recent advances in solving two-stage stochastic (mixed-) integer programs, and have provided some insights and results that exhibit certain interconnections between the methods. Due to the dual angular structure, these solution approaches apply various versions of decomposition, branch-and-bound, and convexification techniques. We have studied these methods from the viewpoint of the adopted decomposition


433

Table 12.1 Literature on solving stochastic mixed-integer programs First Stage (x)

Second Stage (y)

Literature Approach

Section

Binary

Continuous

[23] [3]

L-shaped Branch-and-Fix

12.2.1 12.2.2

Continuous

Integer

[32]

Gr¨ obner basis

12.4

Integer

Integer

[17] [22]

Test-set decomposition 12.2.3 Value function 12.4

Mixed-integer

Integer

[1] [14]

χ-transformation Dual pricing

12.4 12.2.1

Continuous

0—1 mixed-integer [13] [27]

Disj. prog. Disj. prog.

12.3.1 12.3.2

Binary

0—1 mixed-integer [39] [36] [35]

Benders & RLT Disj. prog. Disj. prog.

12.3.1 12.3.1 12.3.1

0—1 mixed-integer 0—1 mixed-integer [41]

Convexification

12.3.2

Mixed-integer

Lagrangian dual Lagrangian dual

12.2.2 12.2.2

Mixed-integer

[12] [33]

framework, the convexification approach used in solving problems having integer recourse, and the enumeration scheme employed when the technology matrix is deterministic. Table 12.1 lists the literature we have covered, grouped by the type of variables appearing in the two stages, which is most relevant to the algorithmic developments. Many of these methods are theoretically extendable to multistage SMIPs. However, scalability becomes a major issue here, which requires further research. For an introduction on multistage stochastic integer programs, we refer the reader to Römisch and Schultz [28]. Acknowledgment This research is supported by the National Science Foundation under Grant Numbers DMI-0552676, DMI-0245643, and DMI-0455807.

References 1. Ahmed, S., Tawarmalani, M., and Sahinidis, N. V. (2004). A finite branch and bound algorithm for two-stage stochastic integer programs. Mathematical Programming 100(2), 355—377. 2. Ahmed, S. and Garcia, R. (2004). Dynamic capacity acquisition and assignment under uncertainty. Annals of Operations Research 124, 267—283. 3. Alonso-Ayuso, A., Escudero, L. F., and Ortu˜ no, M. T. (2003). BFC, a branch-andfix coordination algorithmic framework for solving some types of stochastic pure and mixed 0—1 programs. European Journal of Operational Research 151, 503—519.

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4. Alonso-Ayuso, A., Escudero, L. F., Gar´ın, A., Ortu˜ no, M. T., and P´ erez, G. (2003). An approach for strategic supply chain planning under uncertainty based on stochastic 0—1 programming. Journal of Global Optimization 26, 97—124. 5. Balas, E. (1979). Disjunctive programming. Annals of Discrete Mathematics 5, 3—51. 6. Balas, E. (1997). A modified lift-and-project procedure. Mathematical Programming 79(1—3), 19—31. 7. Balas, E., Ceria, S., and Cornu´ ejols, G. (1993). A lift-and-project cutting plane algorithm for mixed 0—1 programs. Mathematical Programming 58, 295—324. 8. Balas, E., Ceria, S., and Cornuéjols, G. (1996). Mixed 0—1 programming by lift-andproject in a branch-and-cut framework. Management Science 42(9), 1229—1246. 9. Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4, 238—252. 10. Blair, C. and Jeroslow, R. (1978). A converse for disjunctive constraints. Journal of Optimization Theory and Applications 25, 195—206. 11. Blair, C. and Jeroslow, R. (1982). The value function of an integer program. Mathematical Programming 23, 237—273. 12. Carøe, C. C. and Schultz, R. (1999). Dual decomposition in stochastic integer programming. Operations Research Letters 24(1—2), 37—45. 13. Carøe, C. C. and Tind, J. (1997). A cutting-plane approach to mixed 0—1 stochastic integer programs. European Journal of Operational Research 101(2), 306—316. 14. Carøe, C. C. and Tind, J. (1998). L-shaped decomposition of two-stage stochastic programs with integer recourse. Mathematical Programming 83a(3), 451—464. 15. Graver, J. E. (1975). On the foundation of linear and integer programming I. Mathematical Programming 9, 207—226. 16. Hemmecke, R. (2000). On the positive sum property of Graver test sets. Preprint SMDU-468, University of Duisburg, http://www.uni-duisburg.de/FB11/disma/ramon/ articles/preprint2.ps. 17. Hemmecke, R. and Schultz, R. (2003). Decomposition of test sets in stochastic integer programming. Mathematical Programming, Ser. B 94, 323—341. 18. Jeroslow, R. G. (1980). A cutting plane game for facial disjunctive programs. SIAM Journal of Control and Optimization 18, 264—280. 19. Klein Haneveld, W. K., Stougie, L., and van der Vlerk, M. H. (1995) On the convexhull of the simple integer recourse objective function. Annals of Operations Research 56, 209—224. 20. Klein Haneveld, W. K., Stougie, L., and van der Vlerk, M. H. (1996) An algorithm for the construction of convex hulls in simple integer recourse programming. Annals of Operations Research 64, 67—81. 21. Klein Haneveld, W. K. and van der Vlerk, M. H. (1999). Stochastic integer programming: general models and algorithms. Annals of Operations Research 85, 39—57. 22. Kong, N., Schaefer, A. J., and Hunsaker, B. (2006). Two-stage integer programs with stochastic right-hand sides: a superadditive dual approach. Mathematical Programming, 108 (2—3), 275—296. 23. Laporte, G. and Louveaux, F. V. (1993). The integer L-shaped method for stochastic integer programs with complete recourse. Operations Research Letters 13(3), 133—142. 24. Martin, R. K. (1998). Large Scale Linear and Integer Optimization: A Unified Approach, Kluwer Academic, Norwell, MA. 25. Nemhauser, G. L. and Wolsey, L. A. (1999). Integer and Combinatorial Optimization, Wiley-Interscience, New York, 2nd edition. 26. Ntaimo, L. and Sen, S. (2005). The million-variable “march” for stochastic combinatorial optimization. Journal of Global Optimization 32(3), 385—400. 27. Ntaimo, L. and Sen, S. (2008). A branch-and-cut algorithm for two-stage stochastic mixed-binary programs with continuous first-stage variables. International Journal of Computational Science and Engineering 3(6), 232—241.


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28. R¨ omisch, W. and Schultz, R. (2001). Multi-stage stochastic integer programs: An introduction. In: M. Gr¨ otschel, S. O. Krumke, and J. Rambau (ed), Online Optimization of Large Scale Systems, 581—600. Springer, Berlin. 29. Schultz, R. (1995). On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Mathematical Programming 70(1), 73—89. 30. Schultz, R. (2003). Stochastic programming with integer variables. Mathematical Programming 97(1—2), 285—309. 31. Schultz, R., Stougie, L., and van der Vlerk, M. H. (1996). Two-stage stochastic integer programming: a survey. Statistica Neerlandica 50(3), 404—416. 32. Schultz, R., Stougie, L., and van der Vlerk, M. H. (1998). Solving stochastic programs with integer recourse by enumeration: A framework using Gr¨ obner basis reductions. Mathematical Programming 83(2), 229—252. 33. Schultz, R. and Tiedemann, S. (2003). Risk aversion via excess probabilities in stochastic programs with mixed-integer recourse. SIAM Journal on Optimization 14(1), 115—138. 34. Sen, S. (2005). Algorithms for stochastic mixed-integer programming models. In: K. Aardal, G. Nemhauser, and R. Weismantel (ed), Handbooks in Operations Research and Management Science: Discrete Optimization, Vol. 12, Chapter 9, 511—558. Elsevier, Dordrecht. 35. Sen, S. and Higle, J. L. (2005). The C 3 theorem and a D2 algorithm for large scale stochastic mixed-integer programming: Set convexification. Mathematical Programming 104(1), 1—20. 36. Sen, S. and Sherali, H. D. (2006). Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Mathematical Programming 106(2), 203—223. 37. Sherali, H. D. and Adams, W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero—one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411—430. 38. Sherali, H. D. and Adams, W. P. (1994). A hierarchy of relaxations and convex hull characterizations for mixed-integer zero—one programming problems. Discrete Applied Mathematics 52(1), 83—106. 39. Sherali, H. D. and Fraticelli, B. M. P. (2002). A modification of Benders’ decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse. Journal of Global Optimization 22, 319—342. 40. Sherali, H. D. and Shetty, C. M. (1980). Optimization with Disjunctive Constraints. Lecture Notes in Economics and Mathematical Systems, Vol. 181. Springer-Verlag, Berlin. 41. Sherali, H. D. and Zhu, X. (2006). On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables. Mathematical Programming, 108 (2—3), 597—616. 42. Stougie, L. and van der Vlerk, M. H. (1997). Stochastic integer programming. In: M. Dell’Amico, F. Maffioli, and S. Martello (ed), Annotated Bibliographies in Combinatorial Optimization, 127—141. Wiley, Chichester. 43. Van Slyke, R. and Wets, R. (1969). L-shaped linear programs with applications to control and stochastic programming. SIAM Journal on Applied Mathematics 17, 638—663. 44. Wollmer, R. M. (1980). Two-stage linear programming under uncertainty with 0—1 first-stage variables. Mathematical Programming 19, 279—288.

Chapter 13

Dualistic Riemannian Manifold Structure Induced from Convex Functions Jun Zhang and Hiroshi Matsuzoe

Key words: Legendre—Fenchel duality, biorthogonal coordinates, Riemannian metric, conjugate connections, equiaffine geometry, parallel volume form, affine immersion, Hessian geometry

13.1 Introduction Convex analysis has wide applications in science and engineering, such as mechanics, optimization and control, theoretical statistics, mathematical economics and game theory, and so on. It offers an analytic framework to treat systems and phenomena that depart from linearity, based on an elegant mathematical characterization of the notion of “duality” (Rockafellar, 1970, 1974, Ekeland and Temam, 1976). Recent work of David Gao (2000) further provided a comprehensive and unified treatment of duality principles in convex and nonconvex systems, greatly enriching the theoretical foundation and scope of applications. Central to convex analysis is the Legendre—Fenchel transform, and duality between two sets of variables defined on a pair of vector spaces that are dual with respect to each other. When the convex functions involved are smooth, these variables are in one-to-one correspondence; they can actually be viewed as two coordinate systems on a certain Riemannian manifold. This is the viewpoint from the so-called information geometry (Amari, 1985, Amari and Nagaoka, 2000), and it is investigated at great length in this chapter. Jun Zhang · Department of Psychology, University of Michigan, Ann Arbor, MI 48109, U.S.A., e-mail: [email protected] Hiroshi Matsuzoe · Department of Computer Science and Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_13, © Springer Science+Business Media, LLC 2009

437

438

J. Zhang, H. Matsuzoe

The link between convex functions and Riemannian geometry is shown to be severalfold. First, the pair of convex functions conjugate to one another are the potential functions that induce the Riemannian metric. Second, the two sets of variables are special coordinate systems of the manifold in that they are “biorthogonal;” that is, the Jacobian of coordinate transformation between them is precisely the Riemannian metric. It turns out that biorthogonal coordinates are global coordinates for a pair of dually flat connections on the Riemannian manifold. Third, the Fenchel inequality provides a natural way to construct directed (“pseudo-”) distance over the convex point set; this is the Bregman divergence (a.k.a. canonical divergence), which gives rise to the dually flat connections. Finally, the geometric structure (Riemannian metric, conjugate/dual connections) can be induced from graph immersions of a convex function into a higher-dimensional affine space. Our goal in this chapter is to review such a geometric view of convex functions and the associated conjugacy/duality, as well as provide some new results. We review the background of convex analysis and Riemannian geometry (and affine hypersurface theory) in Section 13.2, with attention to the well-established relation between biorthogonal coordinates and dually flat (also called “Hessian”) manifolds. In Section 13.3, we develop the fullfledged α-Hessian geometry, which extends the dually flat Hessian manifold (α = ±1), and give an example from theoretical statistics when such geometry arises; this parallels the generalization of the convex-induced divergence function with arbitrary α (Zhang, 2004) from Bregman divergence (α = ±1). To close, we give a summary and discuss some open problems in Section 13.4.

13.2 Convex Functions and Riemannian Geometry 13.2.1 Convex Functions and the Associated Divergence Functions A strictly convex (or simply “convex”) function Φ : V ⊆ Rn → R, x → 7 Φ(x) is defined by µ ¶ 1+α 1−α 1+α 1−α Φ(x) + Φ(y) − Φ x+ y >0 (13.1) 2 2 2 2 for all x 6= y for any |α| < 1 (the inequality sign is reversed when |α| > 1). In this chapter, V (and Ve below) identifies a subset of Rn both as a point set and as a vector space. We assume Φ to be sufficiently smooth (differentiable up to fourth order). Define BΦ (x, y) = Φ(x) − Φ(y) − hx − y, ∂Φ(y)i,

(13.2)

13 Dualistic Geometry from Convex Functions

439

where ∂Φ = [∂1 Φ, . . . , ∂n Φ] with ∂i ≡ ∂/∂xi denotes the gradient valued in the co-vector space Ve ⊆ Rn , and h·, ·in denotes the canonical pairing of a point/vector x = [x1 , . . . , xn ] ∈ V and a point/co-vector u = [u1 , . . . , un ] ∈ Ve (dual to V ): n X hx, uin = xi ui . (13.3) i=1

(Where there is no danger of confusion, the subscript n in h·, ·in is often omitted.) A basic fact in convex analysis is that the necessary and sufficient condition for a smooth function Φ to be convex is BΦ (x, y) > 0

(13.4)

for x 6= y. We remark that BΦ is sometimes called “Bregman divergence” (Bregman, 1967), widely used in convex optimization literature (Della Pietra et al., 2002, Bauschke, 2003, Bauschke and Combettes, 2003, Bauschke et al., 2003). Zhang (2004) introduced the following family of functions on V × V as indexed by α ∈ R, µ µ ¶¶ 4 1−α 1+α 1−α 1+α (α) DΦ (x, y) = Φ(x) + Φ(y) − Φ x+ y . 1 − α2 2 2 2 2 (13.5) (±1) Here DΦ (x, y) is defined by taking limα→±1 : (1)

(−1)

DΦ (x, y) = DΦ

(−1) DΦ (x, y)

=

(y, x) = BΦ (x, y),

(1) DΦ (y, x)

= BΦ (y, x).

(α)

Note that DΦ (x, y) satisfies the relation (called “referential duality” in Zhang, 2006a) (α) (−α) DΦ (x, y) = DΦ (y, x); that is, exchanging the asymmetric status of the two points (in the directed distance) amounts to α ↔ −α. (α) From its construction, DΦ (x, y) is nonnegative for |α| < 1 due to equation (13.1), and for |α| = 1 due to equation (13.4). For |α| > 1, assuming (α) (((1 − α) /2) x + ((1 + α) /2) y) ∈ V , the nonnegativity of DΦ (x, y) can also be proven due to the inequality (13.1) reversing its sign. Therefore, we have Lemma 13.1. For a smooth function Φ : V ⊆ Rn → R, the following conditions are equivalent (for x, y ∈ V ). (i) Φ is strictly convex. (1) (ii) DΦ (x, y) ≥ 0. (−1) (iii) DΦ (x, y) ≥ 0.

440

J. Zhang, H. Matsuzoe (α)

(iv) DΦ (x, y) ≥ 0 for all |α| < 1. (α) (v) DΦ (x, y) ≥ 0 for all |α| > 1. e : Ve ⊆ Rn → R is Recall that, when Φ is convex, its convex conjugate Φ defined through the Legendre—Fenchel transform: e Φ(u) = h(∂Φ)−1 (u), ui − Φ((∂Φ)−1 (u)),

(13.6)

e e is also convex, and through e e −1 . The function Φ with Φ = Φ and (∂Φ) = (∂ Φ) which (13.4) precisely expresses the Fenchel inequality e Φ(x) + Φ(u) − hx, ui ≥ 0

for any x ∈ V , u ∈ Ve , with equality holding if and only if

e −1 (x) ←→ x = (∂ Φ)(u) e u = (∂Φ)(x) = (∂ Φ) = (∂Φ)−1 (u),

(13.7)

or, in component form,

ui =

e ∂Φ ∂Φ i ←→ x = . ∂xi ∂ui

(13.8)

With the aid of conjugate variables, we can introduce the “canonical divergence” AΦ : V × Ve → R+ (and AΦe : Ve × V → R+ ) where R+ = R+ ∪ {0} e − hx, vi = A e (v, x). AΦ (x, v) = Φ(x) + Φ(v) Φ

They are related to the Bregman divergence (13.2) via

e v). BΦ (x, (∂Φ)−1 (v)) = AΦ (x, v) = BΦe ((∂ Φ)(x),

Bregman (or canonical) divergence1 provides a measure of directed distance between two points; that is, it is nonnegative for all values of x, y ∈ V , and vanishes only when x = y. More formally, a divergence function D : V × V → R+ is a smooth function (differentiable up to third order) that satisfies (i) D(x, y) ≥ 0 ∀x, y ∈ V with equality holding if and only if x = y, ¯ (ii) ∂xi D(x, y)|x=y = ∂yi D(x, y)¯x=y = 0, ¯ (iii) ∂xi ∂yj D(x, y)¯x=y is negative definite.

Here ∂xi denotes partial derivative with respect to the ith component of the x-variable only.2 1 The divergence function, also called the “contrast function,” is a terminology arising out of the theoretical statistics literature. It has nothing to do with the divergence operation in vector calculus. 2 The reader should not confuse the shorthand notations ∂ with ∂ i xi (or ∂y i ): the former operates on a function defined on V such as Φ : x 7→ Φ(x) ∈ R, whereas the latter operates on a function defined on V × V such as D : (x, y) 7→ D(x, y) ∈ R+ .


441

13.2.2 Differentiable Manifold: Metric and Connection Structures A differentiable manifold M is a space that locally “looks like” a Euclidean space Rn . By “looks like,” we mean that for any base (reference) point p ∈ M, there exists a bijective mapping (“coordinate functions”) between the neighborhood of p (i.e., a patch of the manifold) and a subset V of Rn . By locally, we mean that various such mappings must be smoothly related to one another (if they are centered at the same reference point) or consistently glued together (if they are centered at different reference points) and globally cover the entire manifold. Below, we assume that a coordinate system is chosen such that each point is indexed by x ∈ V , with the origin as the reference point. A manifold is specified with certain structures. First, there is an innerproduct structure associated with tangent spaces of the manifold. This is given by the metric tensor field g which is, when evaluated at each location x (omitted in our notation), a symmetric bilinear form g(·, ·) of tangent vectors X, Y ∈ Tp (M) ' Rn such that g(X, X) is always positive for all nonzero i vectors X. In local coordinates P iwith bases P ∂i i≡ ∂/∂x , i = 1, . . . , n (i.e., X, Y are expressed as X = i X ∂i , Y = i Y ∂i ), the components of g are denoted as gij (x) = g(∂i , ∂j ). (13.9) The metric tensor allows us to define distance on a manifold as the shortest curve (called “geodesic”) connecting two points. It also allows the measurement of angles and hence defines orthogonality of a vector to a submanifold. Projections of vectors to a lower-dimensional submanifold become possible once a metric is given. Second, there is a structure associated with the notion of “parallelism” of vector fields on a manifold. This is given by the affine (linear) connection (or simply “connection”) ∇, mapping two vector fields X and Y to a third one denoted by ∇Y X : (X, Y ) 7→ ∇Y X. Intuitively, it represents the “intrinsic” difference of the vector field X from its value at point x and its value at a nearby point connected to x (in the direction given by Y ). Here “intrinsic” means that vector comparison at two neighboring locations of the manifold is through a process called “parallel transport,” whereby a vector’s components are adjusted as it moves across points on the base manifold. Under the local coordinate system with bases ∂i ≡ ∂/∂xi , components of ∇ can be written out in “contravariant” form denoted Γijl (which is a collection of n3 functions of x), X ∇∂i ∂j = Γijl ∂l . (13.10) l

Under coordinate transform x 7→ x ˜, the new set of functions Γe are related to old ones Γ via

442


⎛ ⎞ 2 k X X ∂xi ∂xj ∂ ∂x ˜l x l ⎝ Γemn (˜ x) = Γijk (x) + m n ⎠ k . m n ∂x ˜ ∂x ˜ ∂x ˜ ∂x ˜ ∂x i,j

(13.11)

k

A curve whose tangent vectors are intrinsically parallel along it is called an “auto-parallel curve.” As a primitive on a manifold, affine connections can be characterized in terms of their torsion and curvature. The torsion T of a connection Γ , which is a tensor itself, isP given by the asymmetric part of the connection T (∂i , ∂j ) = ∇∂i ∂j − ∇∂j ∂i = k Tijk ∂k , where Tijk is its local representation3 given as k Tijk (x) = Γijk (x) − Γji (x).

The curviness/flatness of a connection Γ is described by the curvature tensor R, defined as R(∂i , ∂j )∂k = (∇∂i ∇∂j − ∇∂j ∇∂i )∂k . P l Writing R(∂i , ∂j )∂k = l Rkij ∂l and substituting (13.10), the components of 4 the curvature tensor are l Rkij (x) =

l l X ∂Γjk (x) ∂Γik (x) X l m l m − + Γim (x)Γjk (x) − Γjm (x)Γik (x). i j ∂x ∂x m m

l By definition, Rkij is antisymmetric when i ↔ j. A connection is said to be l flat when Rkij (x) ≡ 0. Note that this is a tensorial condition, so that the flatness of a connection ∇ is a coordinate-independent property even though the local expression of the connection (in terms of Γ ) is highly coordinatedependent. For any flat connection, there exists a local coordinate system under which Γijk (x) ≡ 0 in a neighborhood; this is the affine coordinate for a flat connection. In the above discussions, metric and connections are treated as inducing separate structures on a manifold. On a manifold where both are defined, then it is convenient to express a connection Γ in its “covariant” form X Γij,k = g(∇∂i ∂j , ∂k ) = glk Γijl . (13.12) l

Although Γijk is the more primitive quantity that does not involve the metric, Γij,k represents the projection of an intrinsically differentiated vector field onto the manifold spanned by the bases ∂k . The covariant form of the curva3

Here and below, we restrict to holonomic coordinate systems in Rn only, where all coordinate bases commute [∂i , ∂j ] = 0 for i 6= j. 4 This componentwise notation of curvature tensor here follows standard differential geometry textbooks, such as Nomizu and Sasaki (1994). On the other hand, information P l geometers, such as Amari and Nagaoka (2000), adopt the notation R(∂i , ∂j )∂k = l Rijk ∂l , P with Rijkl = l Rm ijk gml .


443

ture tensor is (cf. footnote 4) Rlkij =

X

m glm Rkij .

m

When the connection is torsion free, Rlkij is antisymmetric when i ↔ j or when k ↔ l, and symmetric when (i, j) ↔ (l, k). It is P related to the Ricci tensor Ric (to be defined in (13.27) below) via Rickj = i,l Rlkij g il .

13.2.3 Dualistic Structure on a Manifold: Compatibility Between Metric and Connection A fundamental theorem of Riemannian geometry states that given a metric, there is a unique connection (among the class of torsion-free connections) that “preserves” the metric; that is, the following condition is satisfied: b ∂ ∂i , ∂j ) + g(∂i , ∇ b ∂ ∂j ). ∂k g(∂i , ∂j ) = g(∇ k k

(13.13)

b is known as the Levi-Civita connection. Its Such a connection, denoted as ∇, component forms, called Christoffel symbols, are determined by the components of the metric tensor as (“Christoffel symbols of the second kind”) ¶ X g kl µ ∂gil ∂gjl ∂gij k b Γij = + − 2 ∂xj ∂xi ∂xl l

and (“Christoffel symbols of the first kind”) ¶ µ 1 ∂gik ∂gjk ∂gij b Γij,k = + − . 2 ∂xj ∂xi ∂xk

The Levi-Civita connection is compatible with the metric, in the sense that it treats tangent vectors of the shortest curves on a manifold as being parallel (or equivalently, auto-parallel curves are also geodesics). It turns out that one can define a kind of “compatibility” relation more general than expressed by (13.13), by introducing the notion of “conjugacy” (denoted by ∗) between two connections. A connection ∇∗ is said to be “conjugate” to ∇ with respect to g if ∂k g(∂i , ∂j ) = g(∇∂k ∂i , ∂j ) + g(∂i , ∇∗∂k ∂j).

(13.14)

b which satisfies (13.13), is special in the Clearly, (∇∗ )∗ = ∇. Moreover, ∇, b b ∗ = ∇. sense that it is self-conjugate (∇) Because metric tensor g provides a one-to-one mapping between points in the tangent space (i.e., vectors) and points in the cotangent space (i.e.,

444


co-vectors), (13.14) can also be seen as characterizing how co-vector fields are to be parallel-transported in order to preserve their dual pairing h·, ·i with vector fields. Writing out (13.14) explicitly, ∂gij ∗ = Γki,j + Γkj,i , ∂xk

(13.15)

where analogous to (13.10) and (13.12), X Γij∗l ∂l ∇∗∂i ∂j = l

so that ∗ Γkj,i = g(∇∗∂j ∂k , ∂i ) =

X

∗l gil Γkj .

l

In the following, a manifold M with a metric g and a pair of conjugate connections Γ , Γ ∗ with respect to g is called a “Riemannian manifold with dualistic structure,” and denoted by {M, g, Γ, Γ ∗ }. Obviously, Γ and Γ ∗ satisfy the relation (in either covariant or contravariant forms) 1 Γb = (Γ + Γ ∗ ). 2

More generally, in information geometry, a one-parameter family of affine connections Γ (α) , called “α-connections” (α ∈ R), is introduced (Amari, 1985, Amari and Nagaoka, 2000) Γ (α) =

1+α 1−α ∗ Γ+ Γ . 2 2

(13.16)

Obviously, Γ (0) = Γb. ∗ It can be shown that the curvatures Rlkij , Rlkij for the pair of conjugate ∗ connections Γ , Γ satisfy ∗ Rlkij = Rlkij . So, Γ is flat if and only if Γ ∗ is flat. In this case, the manifold is said to be “dually flat.” When Γ , Γ ∗ are dually flat, then Γ (α) is called “α-transitively flat” (Uohashi, 2002). In such case, {M, g, Γ (α) , Γ (−α) } is called an “α-Hessian manifold,” or a manifold with α-Hessian structure.

13.2.4 Biorthogonal Coordinate Transformation Consider coordinate transform x 7→ u,


∂i ≡

445

X ∂xl ∂ X ∂ = = J li ∂l , ∂ui ∂ui ∂xl l

l

where the Jacobian matrix J is given by Jij (x) =

∂ui , ∂xj

J ij (u) =

∂xi , ∂uj

X

Jil J lj = δij ,

(13.17)

l

where δij is the Kronecker delta (taking the value of 1 when i = j and 0 otherwise). If the new coordinate system u = [u1 , . . . , un ] (with components expressed by subscripts) is such that Jij (x) = gij (x),

(13.18)

then the x-coordinate system and the u-coordinate system are said to be “biorthogonal” to each other because, from the definition of a metric tensor (13.9), X X X g(∂i , ∂ j ) = g(∂i , J lj ∂l ) = J lj g(∂i , ∂l ) = J lj gil = δij . l

l

l

In such a case, denote g ij (u) = g(∂ i , ∂ j ),

(13.19)

which equals J ij (u), the Jacobian of the inverse coordinate transform u 7→ x. Also introduce the (contravariant version) of the affine connection Γ under the u-coordinate system and denote it by an unconventional notation Γtrs defined by X ∇∂ r ∂ s = Γtrs ∂ t ; t

similarly

Γt∗rs

is defined via

∇∗∂ r ∂ s =

X

Γt∗rs ∂ t .

t

The covariant version of the affine connections is denoted by superscripted Γ and Γ ∗ : Γ ij,k (u) = g(∇∂ i ∂ j , ∂ k ),

Γ ∗ij,k (u) = g(∇∗∂ i ∂ j , ∂ k ).

(13.20)

As in (13.11), the affine connections in u-coordinates (expressed in superscript) and in x-coordinates (expressed in subscript) are related via ⎛ ⎞ 2 k X X ∂xr ∂xs ∂ x ⎝ ⎠ ∂uk Γtrs (u) = Γijk (x) + (13.21) ∂u ∂u ∂u ∂u ∂xt i j r s i,j k

446


and Γ rs,t (u) =

X ∂xr ∂xs ∂xt ∂ 2 xt Γij,k (x) + . ∂ui ∂uj ∂uk ∂ur ∂us

(13.22)

i,j,k

Similar relations hold between Γt∗rs (u) and Γij∗k (x), and between Γ ∗rs,t (u) ∗ and Γij,k (x). Analogous to (13.15), we have the following identity, ∂g rt (u) ∂ 2 xt = = Γ rs,t (u) + Γ ∗ts,r (u), ∂us ∂ur ∂us which leads to Proposition 13.1. Under biorthogonal coordinates, the component forms of the metric tensor satisfy X g ik (u)gkj (x) = δji k

while the pair of conjugate connections Γ , Γ ∗ satisfies X Γ ∗ts,r (u) = − g ir (u)g js (u)g kt (u)Γij,k (x)

(13.23)

i,j,k

and Γr∗ ts (u) = −

X

t g js (u)Γjr (x).

(13.24)

j

Next, we discuss the conditions under which biorthogonal coordinates exist on an arbitrary Riemannian manifold. From its definition (13.18), we can easily show that Lemma 13.2. A Riemannian manifold M with metric gij admits biorthogonal coordinates if and only if ∂gij /∂xk is totally symmetric,5 ∂gij (x) ∂gik (x) = . k ∂x ∂xj

(13.25)

That (13.25) is satisfied for biorthogonal coordinates is evident by virtue of (13.17) and (13.18). Conversely, given (13.25), there must be n functions ui (x), i = 1, 2, . . . , n such that ∂uj (x) ∂ui (x) = gij (x) = gji (x) = . ∂xj ∂xi ¢ ¡ Note that ∂gij /∂xk ≡ ∂k (g(∂i , ∂j )) 6= (∂k g)(∂i , ∂j ), the latter is necessarily totally symmetric whenever there exist a pair of torsion-free connections Γ , Γ ∗ that are conjugate with respect to g. 5


447

The above identity, in turn, implies that there exists a function Φ such that ui = ∂i Φ and, by positive definiteness of gij , Φ would have to be a strictly convex function! In this case, the x- and u-variables satisfy (13.7), and the e is related to gij and g ij by pair of convex functions, Φ and its conjugate Φ, e ∂ 2 Φ(x) ∂ 2 Φ(u) ←→ g ij (u) = . i j ∂x ∂x ∂ui ∂uj

gij (x) =

It follows from Lemma 13.2 that a necessary and sufficient condition for a Riemannian manifold to admit biorthogonal coordinates is that its Levi-Civita connection is given by ¶ µ 1 ∂gik ∂gjk ∂gij 1 ∂gij Γbij,k (x) ≡ + − . = j i k 2 ∂x ∂x ∂x 2 ∂xk From this, the following can be shown.

Proposition 13.2. A Riemannian manifold {M, g} admits a pair of biorthogonal coordinates x and u if and only if there exists a pair of conjugate connections γ and γ ∗ such that γij,k (x) = 0, γ ∗rs,t (u) = 0. In other words, biorthogonal coordinates are affine coordinates for dually flat conjugate connections. In fact, we can now define a pair of torsion-free connections by γij,k (x) = 0,

∗ γij,k (x) =

∂gij ∂xk

and show that they are conjugate with respect to g; that is, they satisfy (13.14). This is to say that we select an affine connection γ such that x is its affine coordinate. From (13.22), when γ ∗ is expressed in u-coordinates, X

∂xk ∂gij (x) ∂g ts (u) + ∂ut ∂xk ∂ur i,j,k µ ¶ X ∂g js (u) ∂g ts (u) = g ir (u) − gij (x) + ∂ut ∂ur i,j

γ ∗rs,t (u) =

=−

g ir (u)g js (u)

X j

δjr

∂g js (u) ∂g ts (u) + = 0. ∂ut ∂ur

This implies that u is an affine coordinate system with respect to γ ∗ . Therefore, biorthogonal coordinates are affine coordinates for a pair of dually flat connections. Such a manifold {M, g, γ, γ ∗ } is called a “Hessian manifold” (Shima, 2007, Shima and Yagi, 1997). It is a special case of the α-Hessian manifold (introduced in Section 13.3.2).

448


13.2.5 Equiaffine Structure and Parallel Volume Form on a Manifold For a restrictive class of connections, called “equiaffine” connections, the manifold M may admit uniquely a parallel volume form ω(x). Here, a volume form is a skew-symmetric multilinear map from n linearly independent vectors to a nonzero scalar, and “parallel” is in the sense that (∂i ω)(∂1 , . . . , ∂n ) = 0 where (∂i ω)(∂1 , . . . , ∂n ) ≡ (∇∂i ω)(∂1 , . . . , ∂n ) = ∂i (ω(∂1 , . . . , ∂n )) −

n X l=1

ω(. . . , ∇∂i ∂l , . . .).

Applying (13.10), the equiaffine condition becomes Ã ! n n X X k ∂i (ω(∂1 , . . . , ∂n )) = ω ..., Γil ∂k , . . . l=1

=

n X n X

Γilk δkl

k=1

ω(∂1 , . . . , ∂n ) = ω(∂1 , . . . , ∂n )

l=1 k=1

or

n X

Γill

l=1

X l

Γill (x) =

∂ log ω(x) . ∂xi

(13.26)

Whether a connection is equiaffine is related to the so-called Ricci tensor Ric, defined as the contraction of the curvature tensor R, X k Ricij (x) = Rikj (x). (13.27) k

k For a torsion-free connection Γijk = Γji , applying the definition of the curvature tensor R to the above yields Ã Ã ! ! X X ∂ ∂ Γjll (x) − j Γill (x) (13.28) Ricij − Ricji = ∂xi ∂x l l X k = Rkij . k

One immediately sees that the existence of a function ω satisfying (13.26) is equivalent to the right side of (13.28) being identically zero. In other words, the necessary and sufficient condition for a torsion-free connection


449

to be equiaffine P k is that its Ricci tensor is symmetric, Ricij = Ricji , or equivalently, k Rkij = 0. Making use of (13.26), it is easy to show that the parallel volume form of a Levi-Civita connection Γb is given by q q b (u) = det[g ij (u)]. ω b (x) = det[gij (x)] ←→ ω The parallel volume forms ω, ω ∗ associated with Γ and Γ ∗ satisfy (apart from a positive, multiplicative constant) ω(x) ω ∗ (x) = (b ω (x))2 = det[gij (x)], ∗ ω(u) ω (u) = (b ω (u))2 = det[g ij (u)].

(13.29) (13.30)

Let us now consider the parallel volume forms under biorthogonal coordinates. Contracting the indices t with r in (13.24), and invoking (13.26), we obtain ∂ log ω ∗ (u) ∂ log ω(x(u)) ∂ log ω ∗ (u) X ∂xj ∂ log ω(x) + = + = 0. j ∂us ∂us ∂x ∂us ∂us j After integration, ω ∗ (u) ω(x) = const.

(13.31)

ω(u) ω ∗ (x) = const.

(13.32)

From (13.29)—(13.31), The relations (13.31) and (13.32) indicate that the volume forms of the pair of conjugate connections, when expressed in biorthogonal coordinates respectively, are inversely proportional to each other. Note that ω(x) = ω(∂1 , . . . , ∂n ) and ω ∗ (x) = ω ∗ (∂1 , . . . , ∂n ), as skew-symmetric multilinear maps, transform to ω(u) = w(∂ 1 , . . . , ∂ n ) and ω ∗ (u) = ω ∗ (∂ 1 , . . . , ∂ n ) via ω(x) = det[Jij (x)]ω(u) ←→ ω ∗ (x) = det[J ij (u)]ω ∗ (u), where det[Jij (x)] = det[gij (x)] = (det[J ij (u)])−1 = (det[g ij (u)])−1 . When the pair of equiaffine connections Γ , Γ ∗ are further assumed to be dually flat, then the entire family of α-connections Γ (α) given by (13.16) are equiaffine (Takeuchi and Amari, 2005, Matsuzoe et al., 2006, Zhang, 2007). The Γ (α) -parallel volume element ω (α) can be shown to be given by ω (α) = ω (1+α)/2 (ω ∗ )(1−α)/2 . Clearly, ω (α) (x)ω (−α) (x) = det[gij (x)] ←→ ω (α) (u)ω (−α) (u) = det[g ij (u)].

450


13.2.6 Affine Hypersurface Immersion (of Co-Dimension One) We next discuss dualistic geometry from convex functions as related to hypersurfaces in affine space, which is the subject of study in affine differential geometry (Simon et al., 1991, Nomizu and Sasaki, 1994). Let An+1 be the standard affine space of dimension n + 1, and M an n-dimensional manifold immersed into An+1 as a hypersurface with affine coordinates f = [f 1 , . . . , f n+1 ]; that is, f : M → An+1 . Assume that the local coordinate system on M is x = [x1 , . . . , xn ]. Let ξ = [ξ 1 , . . . , ξ n+1 ] be a vector field defined on M that is “transversal,” that is, nowhere tangential to M. Denote the vector space associated with An+1 as V , with dim(V ) = n + 1, and the canonical pairing of V with its dual vector space Ve (with dim(Ve ) = n + 1) as h , in+1 ; see (13.3). The duplet {f, ξ} is called an “affine immersion.” In local coordinates, they can be explicitly written as functions of x: {f (x), ξ(x)}, where f is valued in A and ξ is valued in V . Because the tangent space Tp (M) is spanned by ¸ ¾ ½∙ a ∂f ∂f a , . . . , , a = 1, . . . , n + 1 , ∂x1 ∂xn we may decompose the second derivatives of f as n

a X ∂2f a k ∂f = Γ + hij ξ a ij ∂xi ∂xj ∂xk

(i, j = 1, . . . , n),

(13.33)

k=1

where hij = hji (called “induced bilinear form” or “affine fundamental form”); if f is convex, then hij is positive definite. The set of coefficients Γijk is called the “induced connection” on M, because it is induced by a flat connection on An+1 . Under coordinate transform, these coefficients can be shown to transform according to (13.21). Similarly, decompose the derivative of ξ a as n a X ∂ξ a k ∂f = − S + τi ξ a , (13.34) i ∂xi ∂xk k=1

Sik

where is known as the “affine shape operator,” and τi is a 1-form on M called the “transversal connection form;” when τ = 0 everywhere on M, the affine immersion {f, ξ} is called “equiaffine.” We define a volume form ω on M arising out of the immersion of {f, ξ}, ω(∂1 , . . . , ∂n ) = Det(∂1 f, . . . , ∂n f, ξ), where Det is the determinant form on An+1 , and ∂i f is the vector field ∂i f = [∂i f 1 , . . . , ∂i f n+1 ]. The covariant derivative of ω is given as follows (see Nomizu and Sasaki, 1994):


451

(∇∂i ω)(∂1 , . . . , ∂n ) = τi ω(∂1 , . . . , ∂n ). This implies that the induced volume form ω is parallel with respect to the induced connection ∇ if and only if {f, ξ} is equiaffine: τ = 0. In order to consider the geometry induced from convex functions and biorthogonal coordinates, we consider a special kind of affine immersion called “graph immersion:” f = [x1 , . . . , xn , Φ(x)],

ξ = [0, . . . , 0, 1],

(13.35)

where Φ is some nondegenerate (in particular, convex) function. Applying (13.33), we obtain the induced connection Γijk (x) = 0 and the affine fundamental form hij (x) as the Hessian of Φ, hij (x) =

∂ 2 Φ(x) . ∂xi ∂xj

Thus the geometry of a graph affine immersion coincides with the Hessian geometry induced from a convex function. Because the transversal vector field ξ is parallel along f , from (13.34), obviously M has an equiaffine structure. ˜ mapping M to An+1 We can define the “dual” of graph immersion, {f˜, ξ}, e e and u given by (13.6) as another graph. Here f˜ = [u1 , . . . , un , Φ(u)], with Φ and (13.7), respectively. The transversal vector field ξ˜ = (0, . . . , 0, 1) is valued in Ve , the dual vector space. The affine fundamental form e h is Because of the identity

e ∂ 2 Φ(u) e hij (u) = . ∂ui ∂uj

X ∂xk ∂xl ∂ 2 Φ(x) e ∂ 2 Φ(u) = , ∂ui ∂uj ∂ui ∂uj ∂xk ∂xl k,l

such affine fundamental form transforms as a 0—2 tensor e hij (u) =

X ∂xk ∂xl hkl (x) ∂ui ∂uj k,l

(even though second derivatives in general do not transform in a tensorlike e the fashion). This means that for dual graph immersions {f, ξ} and {fe, ξ}, e induced affine fundamental form is one and the same h = h. The induced objects {M, h, Γ, Γ ∗ } form a Hessian structure (i.e., induced connections are dually flat). More generally, for an arbitrary affine immersion, we can introduce the notion of “co-normal mapping,” defined as ζ : M → Ve as

452


hξ(x), ζ(x)in+1 = 1, h∂i f (x), ζ(x)in+1 = 0 (i = 1, . . . , n);

(13.36) (13.37)

that is, n+1 X a=1

ξ a (x) ζa (x) = 1,

n+1 X a=1

∂f a (x) ζa (x) = 0 ∂xi

(i = 1, . . . , n).

Intuitively, the co-normal map is a uniquely defined “normal” vector of the tangent hyperplane at f (x). (This property comes from (13.37).) The conormal map is not a unit vector; the “length” of the map is normalized by (13.36). Note that the word “length” and “normal” are in quotation marks because no metric has ever been introduced on V or Ve ; normalization is through the pairing operation h·, ·i. When {f, ξ} is equiaffine, then the co-normal map ζ can be viewed as an immersion from M to An+1 (Nomizu and Sasaki, 1994, p. 57). Specifically, ζ(M) is taken to be (the negative of) the positional vector field (with respect to a center point) in addition to being the transversal vector field. In this case {f˜, ζ} = {−ζ, ζ} is an affine immersion, called the “co-normal immersion” of {f, ξ}. We also call {−ζ, ζ} a “centroaffine immersion” because the immersion has a center, with the position vector −ζ (the first element in the duplet) e . . . the induced objects transversal to its image M. We denote by Γe, e h, τe, S, of {−ζ, ζ}. Then we have the following formulae (see Simon et al., 1991); Γekj,i = −Γki,j + ∂k hij , n X e Sik hkj , hij =

(13.38)

(13.39)

k=1

τei = 0, e Sji = δji .

e are mutually conjugate with respect Equation (13.38) implies that ∇ and ∇ to h. Note that Γ and Γe are, respectively, the induced connections when M is immersed into An+1 in two distinct ways, {f, ξ} and {−ζ, ζ}. Suppose that {f, ξ} is a graph affine immersion with respect to some convex function, and {−ζ, ζ} is the co-normal immersion of {f, ξ}. From (13.34), the affine shape operator S of {f, ξ} vanishes. This implies that e h = 0 from (13.39). Thus, although the co-normal map of an equiaffine immersion is a centroaffine hypersurface in An+1 , the co-normal map of graph immersion has its image lie on an affine hyperplane in An+1 . For an affine immersion {f, ξ} and the co-normal immersion {−ζ, ζ}, we define the “geometric divergence” G on any two points on M by


G(x, y) = hf (x) − f (y), ζ(y)in+1 =

453 n+1 X a=1

(f a (x) − f a (y))ζa (y).

For a graph immersion given by (13.35), we can explicitly solve for ζ from (13.36) and (13.37): ζ = [−∂1 Φ, . . . , −∂n Φ, 1]. Therefore, the expression for geometric divergence becomes G(x, y) = −hx − y, (∂Φ)(y)in + Φ(x) − Φ(y) ≡ BΦ (x, y); geometric divergence is nothing but Bregman divergence (13.2), see Kurose (1994) and Matsuzoe (1998).

13.2.7 Centroaffine Immersion of Co-Dimension Two Now we consider affine immersion of M (with dim(M) = n) into a co-dimension two affine space An+2 (rather than the co-dimension one affine space An+1 as discussed in the last section). In this case, in addition to specifying the immersion, denoted by f : M → An+2 , we need to specify two noncollinear vector fields, both “transversal” on M. The vector space is denoted as V with dim(V ) = n + 2; the dual vector space is denoted as Ve with dim(Ve ) = n + 2. To simply the situation, we consider centroaffine immersion such that one of the transversal vector fields is the (negative of the) positional vector −f and the other is, as before, denoted ξ, that is, the affine immersion is denoted as {f, −f, ξ}; the elements are valued in An+2 , V , V , respectively. The second derivatives of f and ξ are decomposed as follows (for i, j = 1, . . . , n; a = 1, . . . , n + 2); n

a X ∂2f a k ∂f = Γ + hij ξ a − tij f a , ij ∂xi ∂xj ∂xk a

k=1 n X

∂ξ =− ∂xi

k=1

Sik

∂f a + τi ξ a − κi f a . ∂xk

As in affine immersion of co-dimension one, we call Γijk the “induced connection,” hij the “affine fundamental form,” τi the “transversal connection form,” and Sik the “affine shape operator.” Below, we assume that h is positive definite (i.e., f is strictly convex) and τ = 0 (the centroaffine immersion is equiaffine). We denote the “dual map” of {f, −f, ξ} as another centroaffine map taking the form of {f˜, −f˜, ζ}, where the elements are valued in An+2 , Ve , Ve , respectively; f˜ and ζ are specified by

454


hf˜(x), ξ(x)in+2 = 1, hζ(x), ξ(x)in+2 = 0, hf˜(x), f (x)in+2 = 0, hζ(x), f (x)in+2 = 1, ˜ hf (x), ∂i f (x)in+2 = 0, hζ(x), ∂i f (x)in+2 = 0 (i = 1, . . . , n), or explicitly n+2 X a=1 n+2 X a=1 n+2 X

n+2 X

fã (x)ξ a (x) = 1,

a=1 n+2 X

fã (x)f a (x) = 0,

a=1 n+2 X

a

∂f fã (x) i (x) = 0, ∂x a=1

ζa (x)ξ a (x) = 0, ζa (x)f a (x) = 1, ζa (x)

a=1

∂f a (x) = 0 ∂xi

(13.40)

(i = 1, . . . , n). (13.41)

Denote the induced objects as Γe, e h, τe, . . . ; we have the following formulae (see Nomizu and Sasaki, 1994, Matsuzoe, 1998); ∂k hij = Γki,j + Γekj,i , e hij = hij , τei = 0.

(13.42)

We remark that (13.42) is different from (13.39) of the co-dimension one case. If a centroaffine immersion {f, −f, ξ} induces {g, Γ } on M, then the dual map {f˜, −f˜, ζ} induces {g, Γe} on M. This implies that the theory of centroaffine immersions of co-dimension two is more useful than that of affine immersions of co-dimension one when we discuss the duality of statistical manifold. Consider the special case of graph immersion (of co-dimension two) {f, −f, ξ}; that is, f = [x1 , . . . , xn , Φ(x), 1],

ξ = [0, . . . , 0, 1, 0],

(13.43)

where Φ(x) is some convex function. If {f, −f, ξ} has other representations, they are centroaffinely congruent (linearly congruent) to (13.43); hence it suffices to consider (13.43). From straightforward calculations, the dual map {f˜, −f˜, ζ} of {f, −f, ξ} takes the form e f˜ = [−u1 , . . . , −un , 1, Φ(u)],

ζ = [0, . . . , 0, 0, 1].

The left side equation in (13.40) then gives −

n X i=1

e xi ui + Φ(x) + Φ(u) = 0,

(13.44)


455

and the left side equation in (13.41) is −ui +

∂Φ (x) = 0. ∂xi

e is the convex conjugate of Φ as in (13.6), and u = [u1 , . . . , un ] is the Thus, Φ conjugate variable as in (13.7). For graph immersion, it is easy to check that Γki,j = 0, Sik = 0, tij = 0, τi = 0, κi = 0 for all indices and hij (x) =

∂ 2 Φ(x) . ∂xi ∂xj

The same is true for induced objects in dual immersion. Just as in the case of equiaffine immersion {f, ξ} of co-dimension one and the associated co-normal map {−ζ, ζ}, we can construct the geometric divergence G on M for centroaffine immersion {f, −f, ξ} of co-dimension two and the associated dual map {f˜, −f˜, ζ}: G(x, y) = hf˜(y), f (x) − f (y)in+2 = hf˜(y), f (x)in+2 =

n+2 X

fã (y)f a (x).

a=1

For graph immersion, we substitute f and f˜ in (13.43) and (13.44) to yield e G(x, y) = −hx, (∂Φ)(y)in + Φ(x) + Φ((∂Φ)(y)) ≡ BΦ (x, y). In both the equiaffine immersion of co-dimension one (discussed in Section 13.2.6) and centroaffine immersion of co-dimension two (discussed here), the notion of geometric divergence is a generalization of the Bregman (canonical) divergence on a dually flat space. Proposition 13.3. (Kurose, 1994, Matsuzoe, 1998) Let Φ be a strictly convex function on Rn . Then geometric divergence G(x, y) : V × V → R induced by the affine immersion of Φ as a graph in An+1 or by the centroaffine immersion of Φ as a graph in An+2 equals the Bregman divergence BΦ (x, y).

13.3 The α-Hessian Structure Associated with Convex-Induced Divergence The discussion at the end of the last section anticipate a close relation between convex functions and the Riemannian structure on a differentiable manifold

456


whose coordinates are the variables of the convex functions. On such a manifold, divergence functions take the role of pseudo-distance functions that are nonnegative but need not be symmetric. That dualistic Riemannian manifold structures can be induced from a divergence function was first demonstrated by S. Eguchi. Lemma 13.3. (Eguchi, 1983, 1992) A divergence function induces a Riemannian metric g and a pair of conjugate connections Γ , Γ ∗ given as ¯ gij (x) = −∂xi ∂yj D(x, y)¯y=x ; (13.45) ¯ Γij,k (x) = −∂xi ∂xj ∂yk D(x, y)¯y=x ; ¯ ∗ (x) = −∂yi ∂yj ∂xk D(x, y)¯y=x . Γij,k

(13.46)

(13.47)

∗ as given above satisfy (13.15). FurIt is easily verifiable that gij , Γij,k , Γij,k thermore, under arbitrary coordinate transform, these quantities behave properly as desired. Equations (13.45)—(13.47) link a divergence function D to the dualistic Riemannian structure {M, g, Γ, Γ ∗ }. Applying Lemma 13.3 to Bregman divergence BΦ (x, y) given by (13.2) yields ∂ 2 Φ(x) gij (x) = ∂xi ∂xj and ∂ 3 Φ(x) ∗ Γij,k (x) = 0, Γij,k (x) = . ∂xi ∂xj ∂xk Calculating their curvature tensors shows the pair of connections are dually flat. It is commonly referred to, in affine geometry literature, as the “Hessian manifold” (see Section 13.2.4), although in the study by Shima (2007), the potential function Φ need not be convex but only semidefinite. In u-coordinates, these geometric quantities can be expressed as

g ij (u) =

e ∂ 2 Φ(u) , ∂ui ∂uj

Γ ∗ ij,k (u) = 0,

Γ ij,k (u) =

e ∂ 3 Φ(u) , ∂ui ∂uj ∂uk

e is the convex conjugate of Φ. Below, this link from convex functions where Φ to Riemannian manifold is explored in greater detail.

13.3.1 The α-Hessian Geometry We start by reviewing a main result from Zhang (2004) linking the divergence (α) function DΦ (x, y) defined in (13.5) and the α-Hessian structure.


457 (α)

(−α) 6

Proposition 13.4. (Zhang, 2004) The manifold {M, gx , Γx , Γx (α) sociated with DΦ (x, y) is given by gij (x) = Φij and (α)

} as-

(13.48)

1−α 1+α ∗(α) Γij,k (x) = (13.49) Φijk , Φijk . 2 2 denote, respectively, second and third partial derivatives of

Γij,k (x) = Here, Φij , Φijk Φ(x)

Φij =

∂ 2 Φ(x) , ∂xi ∂xj

Φijk =

∂ 3 Φ(x) . ∂xi ∂xj ∂xk

Recall that an α-Hessian manifold is equipped with an α-independent metric and a family of α-transitively flat connections Γ (α) (i.e., Γ (α) satisfying (13.16) and Γ (±1) are dually flat). From (13.49), ∗(α)

(−α)

Γij,k = Γij,k , with the Levi-Civita connection given as: 1 Γbij,k (x) = Φijk . 2

Straightforward calculation shows that:

(α)

(−α)

Corollary 13.1. For α-Hessian manifold {M, gx , Γx , Γx

},

(i) The curvature tensor of the α-connection is given by (α)

Rμνij (x) =

1 − α2 X ∗(α) (Φilν Φjkμ − Φilμ Φjkν )Ψ lk = Rijμν (x), 4 l,k

with Ψ ij being the matrix inverse of Φij , (ii) All α-connections are equiaffine, with the α-parallel volume forms (i.e., the volume forms that are parallel under α-connections) given by ω (α) (x) = det[Φij (x)](1−α)/2 . The reader is reminded that the metric and conjugated connections in the forms (13.48) and (13.49) are induced from (13.5). Using the convex conjugate e : Ve → R given by (13.6), we introduce the following family of divergence Φ e (α) (x, y) : V × V → R+ defined by functions D e Φ 6 The subscript in x (or u below) indicates that the x-coordinate system (or u-coordinate system, resp.) is being used. Recall from Section 13.2.4 that under x (u, resp.) local coordinates g and Γ , in component forms, are expressed by lower (upper, resp.) indices.

458


e (α) (x, y) ≡ D(α) ((∂Φ)(x), (∂Φ)(y)). D e e Φ Φ

Explicitly written, this new family of divergence functions is e (α) (x, y) = D e Φ

4 1 − α2

µ

1−α e 1+α e Φ(∂Φ(x)) + Φ(∂Φ(y)) 2 2 µ ¶¶ 1−α 1+α e −Φ ∂Φ(x) + ∂Φ(y) . 2 2

e (α) (x, y) induces the α-Hessian Straightforward calculation shows that D e Φ (−α)

(α)

structure {M, gx , Γx , Γx } where Γ (∓α) are given by (13.49); that is, the pair of α-connections are themselves “conjugate” (in the sense of α ↔ −α) (α) to those induced by DΦ (x, y).

13.3.2 Biorthogonal Coordinates on α-Hessian Manifold If, instead of choosing x = [x1 , . . . , xn ] as the local coordinates for the manifold M, we use its biorthogonal counterpart u = [u1 , . . . , un ] to index points (α) on M. Under this u-coordinate system, the divergence function DΦ between the same two points on M becomes

Explicitly written, e (α) (u, v) = D Φ

e e e (α) (u, v) ≡ D(α) ((∂ Φ)(u), (∂ Φ)(v)). D Φ Φ

4 1 − α2

µ

1+α 1−α Φ((∂Φ)−1 (u)) + Φ((∂Φ)−1 (v)) 2 2 µ ¶¶ 1−α 1+α −1 −1 −Φ . (∂Φ) (u) + (∂Φ) (v) 2 2

Recalling our notation (13.19) and (13.20), we have (α)

(−α)

Corollary 13.2. The α-Hessian manifold {M, gu , Γu , Γu e (α) (u, v) is given by with D Φ Γ (α)ij,k (u) =

eij (u), g ij (u) = Φ

1 + α eijk Φ , 2

Γ ∗(α)ij,k (u) =

1−α e Φijk . 2

} associated (13.50) (13.51)

eij , Φ eijk denote, respectively, second and third partial derivatives of Here, Φ e Φ(u),


459

Table 13.1 Divergence functions and induced geometry Divergence Function

Defined as

(α) DΦ (x, y)

V × V → R+

(α) Φ

D e ((∂Φ)(x), (∂Φ)(y)) V × V → R+ (α) Φ

Ve × Ve → R+

D e (u, v)

(α) e e Ve × Ve → R+ DΦ ((∂ Φ)(u), (∂ Φ)(v))

2e eij (u) = ∂ Φ(u) , Φ ∂ui ∂uj

Induced Geometry n o (α) (−α) Φij , Γx , Γx n o (−α) (α) Φij , Γx , Γx n o eij , Γu(−α) , Γu(α) Φ n o eij , Γu(α) , Γu(−α) Φ

eijk (u) = Φ

e ∂ 3 Φ(u) . ∂ui ∂uj ∂uk

We remark that the same metric (13.50) and the same α-connections (−α)

(13.51) are induced by DΦe

plication of Lemma 13.3.

(α)

(u, v) ≡ DΦe (v, u); this follows as a simple ap-

An application of (13.23) gives rise to the following relations. X (−α) Γ (α)mn,l (u) = − g im (u)g jn (u)g kl (u)Γij,k (x), i,j,k

Γ

∗(α)mn,l

R

(α)klmn

(u) = − (u) =

X

(α)

g im (u)g jn (u)g kl (u)Γij,k (x),

i,j,k

X

(α)

g ik (u)g jl (u)g μm (u)g νn (u)Rijμν (x).

i,j,μ,ν

The volume form associated with Γ (α) is eij (u)](1+α)/2 . ω (α) (u) = det[Φ

e (α) (u, v), as well as D e (α) (x, y) introduced earlier, take When α = ±1, D Φ e Φ the form of Bregman divergence (13.2). In this case, the manifold is dually (±1) flat, with curvature tensor Rijμν (x) = R(±1)klmn (u) = 0. We summarize the relations between the convex-induced divergence functions and the geometry they generate in Table 13.1.

13.3.3 Applications of α-Hessian Geometry Finally, we give an application of the α-Hessian geometry in mathematical statistics. A statistical model is a set of (what we call) ζ-functions ζ 7→ p(ζ), where a ζ-function is an element of some function space B = {p(·) : X →

460


R, p(ζ) > 0} over a σ-finite set X with dominant measure μ. A parametric model Mθ is defined as Mθ = {p(·|θ) ∈ B, θ ∈ V ⊆ Rn }. That is, Mθ forms a smooth manifold with θ as coordinates. One can define divergence functionals to measure the directed distance between two ζ-functions p and q. The most familiar is the Kullback—Leibler divergence. With the aid of a smooth and strictly convex function f : R → R and a strictly increasing function ρ : R → R, one can show that the following is a general form of convex-induced divergence functional. µ ¶ Z 4 1+α 1−α 1+α 1−α f (ρ(p)) + f (ρ(q)) − f ρ(p) + ρ(q) dμ, 1 − α2 X 2 2 2 2 (13.52) since it is nonnegative and equals zero if and only if p(ζ) = q(ζ) almost surely. A parametric model p(·|θ) ∈ Mθ is said to be “ρ-affine” if there exists a set of linearly independent functions λi (ζ) ∈ B such that X ρ(p(ζ|θ)) = θi λi (ζ). i

The parameter θ = [θ1 , . . . , θn ] is called the “natural parameter” of a ρaffine parametric model, and the functions λ1 (ζ), . . . , λn (ζ) are the affine basis functions. Examples of ρ-affine manifold include the so-called “alphaaffine manifolds” (Amari, 1985, Amari and Nagaoka, 2000), where ρ(·) takes on the following form (indexed by β ∈ [−1, 1]), ⎧ log t β = 1, ⎪ ⎨ (β) l (t) = 2 ⎪ ⎩ t(1−β)/2 β ∈ [−1, 1). 1−β When a parametric model is ρ-affine, the function Ã ! Z Z X Φ(θ) = f (ρ(p(ζ|θ))) dμ = f θi λi (ζ) dμ X

X

i

can be shown to be strictly convex. Therefore, the divergence functional in (α) (13.52) takes the form of the divergence function DΦ (θp , θq ) on V × V given by (α)

DΦ (θp , θq ) =

4 1 − α2

µ

1+α 1−α Φ(θp ) + Φ(θq ) 2 2 µ ¶¶ 1−α 1+α −Φ . θp + θq 2 2


461

This is exactly (13.5)! An immediate consequence is that a ρ-affine manifold is the α-Hessian manifold, with metric and affine connections given by Proposition 13.4. For any ζ-function ζ 7→ p(ζ), we now define Z ηi = f 0 (ρ(p(ζ))) λi (ζ) dμ X

such that η = [η1 , . . . , ηn ] ∈ Ve ⊆ Rn . We call η the “expectation parameter” of p(ζ) with respect to the set of (affine basis) functions λ1 (ζ), . . . , λn (ζ). It can be easily verified that for the ρ-affine parametric models, ηi = Define Φ∗ (θ) =

Z

∂Φ(θ) . ∂θi

f˜(f 0 (ρ(p(ζ|θ)))) dμ, X

e where f˜: R → R is the Fenchel conjugate of f ; then Φ(η) ≡ Φ∗ ((∂Φ)−1 (η)) is the Fenchel conjugate of Φ(θ). The pair of convex functions Φ, Φ∗ induces η, θ via: e ∂ Φ(η) ∂Φ(θ) = ηi ←→ = θi . i ∂θ ∂ηi In theoretical statistics, we can call Φ(θ) the generalized cumulant gene erating function (or partition function), and Φ(η) the generalized entropy function. Natural parameter θ and expectation parameter η, which form biorthogonal coordinates, play important roles in statistical inference.

13.4 Summary and Open Problems e that are mutually conjugate, For two smooth, strictly convex functions Φ, Φ e the variables u = ∂Φ(x) and x = ∂ Φ(u) are in one-to-one correspondence. It has been shown in this chapter that such a pair of variables can be viewed as biorthogonal coordinate systems on a Riemannian manifold whose metric is the second derivative of Φ when the x-coordinate system is used (or of e when the u-coordinate system is used). Furthermore, a family of affine Φ connections (indexed by α) can be defined with nonzero curvatures except for α = ±1, the dually flat case (the so-called “Hessian manifold”). Each of these α-connections is equiaffine and admits a parallel volume form, and (α) e (α) ) the entire family is induced from the divergence function DΦ (or D e Φ e associated with any convex function Φ (or Φ).

462


Our analysis revealed that the conjugate (±α)-connections reflect two kinds of duality embodied by the convex-induced divergence function. The first is referential duality related to the choice of the reference and the comparison status for the two points (x versus y) for computing the value of the (α) (−α) divergence DΦ (x, y) = DΦ (y, x). The second is representational duality (α) related to the construction of two families of divergence functions, DΦ (x, y) (α) versus DΦe ((∂Φ)(x), (∂Φ)(y)), using conjugate convex functions (see Table 13.1 in Section 13.3). The geometric quantities expressed in x-coordinates and expressed in u-coordinates are related to each other via Proposition 13.1. When α = ±1, the two members of divergence functions coincide (and become Bregman divergence), so that the two kinds of duality reveal themselves as biduality: (−1)

DΦ

(−1)

(x, y) = DΦe

(1)

(1)

(∂Φ(y), ∂Φ(x)) = DΦe (∂Φ(x), ∂Φ(y)) = DΦ (y, x) ,

which is compactly written in the form of canonical divergence as AΦ (x, v) = AΦe (v, x) .

The relation between convex-induced divergence functions and α-connections is intriguing; that α as a convex mixture parameter coincides with α as indexing the family of connections is remarkable! We know that, in general, there may be many families of divergence functions that could yield the same α-connections. An explicit construction is as follows. Take the families of divergence functions (γ ∈ R, β ∈ [−1, 1]) 1 + β (γ) 1 − β (−γ) DΦ (x, y) + DΦ (x, y) , 2 2 which induce an α-Hessian structure whose metric and conjugate connections are given in the forms (13.48) and (13.49), with α taking the value of βγ. The nonuniqueness of divergence functions giving rise to the family of αconnections invites the question of how to characterize the convex-induced divergence functions from the perspective of α-Hessian geometry. There is reason to believe that such axiomatization is possible because (i) the form of divergence function for the dually flat manifold (α = ±1) is unique, namely, the Bregman divergence BΦ ; (ii) Lemma 13.1 gives that D(α) ≥ 0 if and only if BΦ ≥ 0 for any smooth function Φ. This hints at a deeper connection yet to be understood between convexity of a function and the α-Hessian geometry. Another topic that needs further investigation is with respect to affine hypersurface realization of the α-Hessian manifold. We know that in affine immersion, geometric divergence is a generalization of the canonical divergence of dually flat (i.e., Hessian) manifolds. How to model the nonflat manifold with a general α value remains an open question. In particular, is there a generalization of geometric divergence that mirrors the way a convex-induced


463

(α)

divergence function DΦ generalizes Bregman divergence BΦ (or equivalently, the canonical divergence AΦ )? Finally, how do we extend the above analysis to an infinite-dimensional setting? The use of convex analysis (in particular, Young function and Orlicz space) to model the infinite-dimensional probability manifold yields fruitful insights for understanding difficult topological issues (Pistone and Sempi, 1995). It would thus be a worthwhile effort to extend the notion of biorthogonal coordinates to the infinite-dimensional manifold to study nonparametric information geometry. To this end, it would also be useful to extend the affine hypersurface theory to the infinite-dimensional setting and provide the formulation for co-dimension one affine immersion and co-dimension two centroaffine immersion. Here, affine hypersurfaces are submanifolds (resulting from normalization and positivity constraints on probability density functions; see, e.g., Zhang and Hasto, 2006) of an ambient manifold of unrestricted Banach space functions. Preliminary analyses (Zhang, 2006b) show that such an ambient manifold is flat for all α-connections, α ∈ R. So it provides a natural setting (i.e., affine space) in which probability densities can be embedded as an affine hypersurface. The value of such a viewpoint for statistical inference remains a topic for future exploration.

References Amari, S. (1985). Differential Geometric Methods in Statistics. Lecture Notes in Statistics 28, Springer-Verlag, New York. Reprinted in 1990. Amari, S. and Nagaoka, H. (2000). Method of Information Geometry. AMS Monograph, Oxford University Press. Bauschke, H.H. (2003). Duality for Bregman projections onto translated cones and affine subspaces. J. Approx. Theory 121, 1—12. Bauschke, H.H. and Combettes, P.L. (2003). Iterating Bregman retractions. SIAM J. Optim. 13, 1159—1173. Bauschke, H.H., Borwein, J.M., and Combettes, P.L. (2003). Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596—636. Bregman, L.M. (1967). The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Physics 7, 200—217. Della Pietra, S., Della Pietra, V., and Lafferty, J. (2002). Duality and auxiliary functions for Bregman distances. Technical Report CMU-CS-01-109, School of Computer Science, Carnegie Mellon University. Eguchi, S. (1983). Second order efficiency of minimum contrast estimators in a curved exponential family. Ann. Statistics 11, 793—803. Eguchi, S. (1992). Geometry of minimum contrast. Hiroshima Math. J. 22, 631—647. Ekeland, I. and Temam, R. (1976). Convex Analysis and Variational Problems. SIAM, Amsterdam. Gao, D.Y. (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic, Dordrecht, xviii+454 pp. Kurose, T. (1994). On the divergences of 1-conformally flat statistical manifolds. Tohoku Math. J. 46, 427—433.

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Matsuzoe, H. (1998). On realization of conformally-projectively flat statistical manifolds and the divergences. Hokkaido Math. J. 27, 409—421. Matsuzoe, H., Takeuchi, J., and Amari. S (2006). Equiaffine structures on statistical manifolds and Bayesian statistics. Differential Geom. Appl. 24, 567—578. Nomizu, K. and Sasaki, T. (1994). Affine Differential Geometry — Geometry of Affine Immersions. Cambridge University Press. Pistone, G. and Sempi, C. (1995). An infinite dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Statistics 33, 1543—1561. Rockafellar, R.T. (1970). Convex Analysis. Princeton University Press. Rockafellar, R.T. (1974). Conjugate Duality and Optimization. SIAM, Philadelphia. Shima, H. (2007). The Geometry of Hessian Structures. World Scientific, Singapore. Shima, H. and Yagi, K. (1997). Geometry of Hessian manifolds. Differential Geom. Appl. 7, 277—290. Simon, U., Schwenk-Schellschmidt, A., and Viesel, H. (1991). Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes, Science University of Tokyo. Takeuchi, J. and Amari, S. (2005). α-Parallel prior and its properties. IEEE Trans. Inf. Theory 51, 1011—1023. Uohashi, K. (2002). On α-conformal equivalence of statistical manifolds. J. Geom. 75, 179—184. Zhang, J. (2004). Divergence function, duality, and convex analysis. Neural Comput. 16, 159—195. Zhang, J. (2006a). Referential duality and representational duality in the scaling of multidimensional and infinite-dimensional stimulus space. In: Dzhafarov, E. and Colonius, H. (Eds.) Measurement and Representation of Sensations: Recent Progress in Psychological Theory. Lawrence Erlbaum, Mahwah, NJ. Zhang, J. (2006b). Referential duality and representational duality on statistical manifolds. Proceedings of the Second International Symposium on Information Geometry and Its Applications, Tokyo (pp. 58—67). Zhang, J. (2007). A note on curvature of α-connections on a statistical manifold. Ann. Inst. Statist. Math. 59, 161—170. Zhang, J. and Hasto, P. (2006). Statistical manifold as an affine space: A functional equation approach. J. Math. Psychol. 50, 60—65.

Chapter 14

NMR Quantum Computing Zhigang Zhang, Goong Chen, Zijian Diao, and Philip R. Hemmer

Summary. Quantum computing is at the forefront of scientific and technological research and development of the 21st century. NMR quantum computing is one the most mature technologies for implementing quantum computation. It utilizes the motion of spins of nuclei in custom-designed molecules manipulated by RF pulses. The motion is on a nano- or microscopic scale governed by the Schrödinger equation in quantum mechanics. In this chapter, we explain the basic ideas and principles of NMR quantum computing, including basic atomic physics, NMR quantum gates, and operations. New progress in optically addressed solid-state NMR is expounded. Examples of Shor’s algorithm for factorization of composite integers and the quantum lattice-gas algorithm for the diffusion partial differential equation are also illustrated.

14.1 Nuclear Magnetic Resonance Many chapters in this book are concerned with mathematical problems in mechanics, elasticity, fluid mechanics, materials, and so on, which are on the macroscale. At the other extreme is the study of problems in atoms and molecules, photonics, nanotechnology, and the like, which are of the microor nanoscale governed chiefly by the Schrödinger equation. This area has unZhigang Zhang · Goong Chen Department of Mathematics, Texas A&M University, College Station, TX 77843 e-mail: [email protected], [email protected] Zijian Diao Department of Mathematics, Ohio University - Eastern, St. Clairsville, OH 43950 e-mail: [email protected] P.R. Hemmer Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, e-mail: [email protected] D.Y. Gao, H.D. Sherali, (eds.), Advances in Applied Mathematics and Global Optimization Advances in Mechanics and Mathematics 17, DOI 10.1007/978-0-387-75714-8_14, © Springer Science+Business Media, LLC 2009

465

466

Z. Zhang, G. Chen, Z. Diao, P.R. Hemmer

dergone rapid advancement during the past ten years, in large part due to the stimuli from laser applications, quantum computing and quantum technology, and nanoelectronics. Most of the practitioners in this area are physicists and it appears that this area has not drawn enough attention from mathematicians. Here, we wish to describe one such development, namely, nuclear magnetic resonance (NMR) quantum computing. There already exist many papers on this topic (see, e.g., [18, 19, 16, 56, 58, 109]) written by physicists and computer scientists. Our chapter here describes the same interest, but perhaps from a more mathematical point of view.

14.1.1 Introduction As of today, NMR is the most mature technology for the implementation of quantum computing. Naturally, this area is rife with papers. A good Internet resource for looking up NMR quantum computing references, both old and new, is the U.S. Los Alamos National Laboratory’s Web site http://xxx.lanl.gov/quant-ph. At present, several types of elementary quantum computing devices have been developed, based on AMO (atomic, molecular, and optical) or semiconductor physics and technologies. We may roughly classify them into the following. Atomic – ion and atom traps, cavity QED [13]. Molecular – NMR. Semiconductor – coupled quantum dots [12], silicon (Kane) [59]. Crystal structure – nitrogen-vacancy (NV) diamond. Superconductivity – SQUID. The above classification is not totally rigorous as new types of devices, such as quantum dots, or ion traps embedded in cavity-QED, have emerged which are of a hybrid nature. Also, laser pulse control, which is of an optical nature, seems to be omnipresent. In [3], a total of 12 types of quantum computing proposals have been listed.1 Nevertheless, it is clear that NMR quantum computing belongs to the class of molecular computing where we use molecules as a small computer. The logic bits are the nuclear spins of atoms in custom-designed molecules. Spin flips are achieved through the application of radio-frequency (RF) fields on resonance at the nuclear spin frequencies. The system can be initialized by cooling the system down to the ground state or known low-entropy state, or using a special technology called averaging, especially for liquid NMR working in room temperature. Measurement or readout is carried out by measuring the magnetic induction signal generated by the 1

The additional proposals not listed above but given in [3] are quantum Hall qubits, electrons in liquid helium, and spin spectroscopies.

14 NMR Quantum Computing

467

precessing spin on the receiver coil. Numerous experiments have been successfully tried for different algorithms, mostly using liquid NMR technology. The algorithms tested include Grover’s search algorithm [108, 58, 46, 122], other generalized search algorithms [76], quantum Fourier transforms [26, 114], Shor’s algorithm [111], Deutsch—Jozsa algorithm [15, 74, 27, 84, 24], order finding [107, 100], error correcting code [65], and dense coding [33]. There are also other implementations reported such as the cat-code benchmark [64], information teleportation [87], and quantum system simulation [98]. NMR is an important tool in chemistry which has been in use for the determination of molecular structure and composition of solids, liquid, and gases since the mid-1940s, by research groups in Stanford and MIT independently, led by F. Bloch and E.M. Purcell, both of whom shared the Nobel prize in physics in 1952 for the discovery. There are many excellent monographs on NMR [31, 91, 82]. There are also many other nice Internet Web-site resources offering concise but highly useful information about NMR (cf., e.g., [28, 51, 115]). Let us briefly explain the physics of NMR by following Edwards [28]. The NMR phenomenon is based on the fact that the spin of nuclei of atoms have magnetic properties that can be utilized to yield chemical, physical, and biological information. Through the famous Stern—Gerlach experiment in the earlier development of quantum mechanics, it is known that subatomic particles (protons, neutrons, and electrons) have spins. Nuclei with spins behave as a bar magnet in a magnetic field. In some atoms, for example, 12 C (carbon-12), 16 O (oxygen16), and 32 S (sulphur-32), these spins are paired and cancel each other out so that the nucleus of the atom has no overall spin. However, in many atoms (1 H, 13 C, 31 P, 15 N, 19 F, etc.) the nucleus does possess an overall spin. To determine the spin of a given nucleus one can use the following rules. 1. If the number of neutrons and the number of protons are both even, the nucleus has no spin. 2. If the number of neutrons plus the number of protons is odd, then the nucleus has a half-integer spin (i.e., 1/2, 3/2, 5/2). 3. If the number of neutrons and the number of protons are both odd, then the nucleus has an integer spin (i.e., 1, 2, 3). In quantum mechanical terms, the nuclear magnetic moment of a nucleus can align with an externally applied magnetic field of strength B0 in only 2I +1 ways, either with or against the applied field B0 , where I is the nuclear spin given in (1), (2), and (3) above. For example, for a single nucleus with I = 1/2, only one transition is possible between the two energy levels. The energetically preferred orientation has the magnetic moment aligned parallel with the applied field (spin m = +1/2) and is often denoted as α, whereas the higher-energy anti-parallel orientation (spin m = −1/2) is denoted as β. See Figure 14.1. In NMR quantum computing, these spin-up and spindown quantum states resemble the two binary states 0 and 1 in a classical computer. Such a nuclear spin can serve as a quantum bit, or qubit. The

468


Energy m=-1/2 (β: spin down)

no magnetic field is applied magnetic field is applied

m=1/2 (α: spin up)

Fig. 14.1 Splitting of energy levels of a nucleus with spin quantum number 1/2. Z

B0

Spinning nucleus with angular momentum μ

Fig. 14.2 A magnetic field B0 is applied along the z-axis, causing the spinning nucleus to precess around the applied magnetic field.

rotational axis of the spinning nucleus cannot be oriented exactly parallel (or anti-parallel) with the direction of the applied field B0 (aligned along the z-axis) but must precess (motion similar to a gyroscope) about this field at an angle, with an angular velocity ω0 , given by the expression ω0 = γB0 . The precession rate ω0 is called the Larmor frequency (cf. Figure 14.2). See more discussion of ω0 below. The constant γ is called the magnetogyric ratio. This precession process generates a magnetic field with frequency ω0 . If we irradiate the sample with radio waves (MHz), then the proton can absorb the energy and be promoted to the higher-energy state. This absorption is called resonance because the frequencies of the applied radiation and the precession coincide at that frequency, leading to resonance. There is another technique related to NMR, called electron spin resonance (ESR), that deals with the spins of electrons instead of those of the nuclei. The principles for ESR are nevertheless similar. Quantum entanglement is accomplished through spin—spin coupling from the electronic bonds between the nuclei within the molecule and special RF pulse manipulations. We now examine some fundamentals of atomic physics that are essential in any quantitative study of the manipulation of the quantum behavior of atoms. A complete description of the Hamiltonian (i.e., energy) of an atom contains nine terms as follows [31]; H = Hel + HCF + HLS + HSS + HZe + HHF + HZn + HII + HQ . (14.1)


469

Receiver coil

Magnet coil Sample Transmitter coil Fig. 14.3 Schematic diagram of an NMR apparatus. A sample which has nonzero-spin nuclei is put in a static magnetic field regulated by the current through the magnet coil. A transmitter coil provides the perpendicular field and a receiver coil picks up the signal. We can change the current through the magnet coil or change the frequency of the current in the transmitter coil to reach resonance.

The first three terms have the highest order, called the atomic Hamiltonian. They are the electronic Hamiltonian term, crystal field term, and the spin—orbit interaction term, respectively. The electronic Hamiltonian consists of kinetic energy of all electrons, mvi2 /2 = p2i /2m, and two Coulomb terms: the potential energy of electrons relative to the nuclei, −zn e2 /rni , and the interelectronic repulsion energy, e2 /rij : Hel =

X p2 X zn e2 X e2 i + , − 2m rni r i i,n i>j ij

where rni denote the distance between the ith electron with the nth nucleus, and rij denote the interelectronic distance between the ith and the j th electrons. The term HCF is called the crystal field term. It comes from the interaction between the electron and the electronically charged ions forming the crystal, and is essentially a type of electrical potential energy: V =−

X Qj e i,j

rij

,

where Qj is the ionic charge and rij is the distance from the electron to the ion. Normally, only those ions nearest to the electron are considered. The third in the atomic Hamiltonian is the interaction between spin and orbit: HLS = λL · S,

470


where L and S are the angular momenta of the orbit and spin, respectively, and λ is the coupling constant. In this section, we use S for the electron spin and I for the nuclear spin. The remaining six terms are called spin Hamiltonians. Terms HZe and HZn are two that result from the application of an external magnetic field: HZe = βB · (L + S), X HZn = − gni βn B · Ii , i

where B is the magnetic field strength. These two terms are called Zeeman terms, and they play major roles in NMR and ESR. The nuclear spin—spin interaction term HII is also important in quantum computation: X HII = Ii · Jij · Ij , i>j

because it provides a mechanism for the interaction between qubits. Hyperfine interaction arises from the interaction between the nuclear magnetic moments and the electron: X HHF = S · Ai · Ii . i

In (14.1), by letting the z-axis be the privileged direction of spin measurement, the spin—spin interaction term HSS is expressed as 1 HSS = D[Sz2 − S(S + 1)] + E(Sx2 − Sy2 ). 3

The very last term in (14.1) is called the quadrupolar energy: µ 2 ¶ ∂ V e2 Q HQ = (3Iz2 − I(I + 1) + η(Ix2 − Iy2 )). 4I(2I − 1) ∂Z 2 For a specific system, only the Hamiltonian playing major roles is needed in the final model. For example, in the study of ESR, only three terms are retained and the Hamiltonian is written as H = HZe + HHF + HSS , whereas in the NMR case, H = HZn + HII .


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14.1.2 More about the Hamiltonian of NMR A classical way to explain NMR is to regard it as a rotating charged particle that acts as a current circulating in a loop ([31, 10]), which creates a magnet with magnetic moment μ, μ = qvr/2, where q is the electronic charge. The particle is rotating at v/2πr revolutions per second. Converting μ to electromagnetic units by dividing it by the velocity of light, and using angular momentum of the particle rather than the velocity of the particle, we obtain μ = (q/2M c)p, where p is the angular momentum oriented along the rotating axis. The ratio μ/p is called the magnetogyric ratio, denoted by γ. A static magnetic field with strength B will apply a torque, which is equal to μ × B, on this particle. Newton’s law states that the angular momentum will change according to a differential equation dp q =μ×B= p × B. dt 2M c Computation shows that p will rotate around the direction of B with frequency ω0 defined by q ω0 = B. 2M c The above is called the Larmor equation, and the frequency ω0 is called the Larmor frequency, the precession frequency, or the resonance frequency as mentioned previously in Figure 14.2. The above classical considerations are now modified by quantization to incorporate the quantum-mechanical behaviors of the nuclear spin. The vector variable p is quantized with quantum number (I(I +1))1/2 , and its projection to the z-axis (the direction of the magnetic field) is m~. In total, there are 2I + 1 valid values of m evenly distributed from −I to I; that is, m = −I, −I + 1, . . . , I − 1, I. A factor g is introduced to include both the spin and orbital motion in the total angular momentum, called the Landé or spectroscopic splitting factor. For a free electron and proton, the magnetic momenta can be given as µ ¶ ge he ge β μe = = , 2 4πMe c 2 µ ¶ he μn = gn I = gn I βN , 4πMN c where ge = 2.0023, gn = 5.58490. Numbers β and βN are called, respectively, the Bohr and the nucleus magneton where β = 9.27 × 10−21 erg gauss−1 and βN = 5.09 × 10−24 erg gauss−1 . These values vary for different particles. In NMR, it is convenient to use the resonance frequency ω0 :

472


~ω0 = ge βB0 , ~ω0 = gN I βN B0 . Now we can write the Hamiltonian of a free nucleus as H = −μ · B = −~γI · B,

(14.2)

where γ is the magnetogyric ratio defined by γ = μ/I~ just as in the classical case. It is a characteristic constant for every type of nuclei; different nuclei have different magnetogyric ratios. Vector I after quantization, becomes the operator of angular momentum. The eigenvalues of this system, or the energy levels are E = γ~mB, m = −I, −I + 1, . . . , I − 1, I. (14.3) The difference between two neighboring energy levels is γ~B, which defines the resonance frequency depending on the magnetic field B and the particle. There are other factors to be considered. The resonance frequency changes with the chemical environment of the nucleus. An example is the fluorine resonance spectrum of perfluorioisopropyl iodide. Two resonance lines of fluorine are observed in the spectrum, and the intensities ratio 6:1 agrees with the population ratio of the two groups of fluorine atoms. This phenomenon, called the chemical shift, is proportional to the strength of the magnetic field applied. This effect comes up because electrons close to the nucleus change the magnetic field around it; in other words, they create a diamagnetic shielding surrounding the nucleus. If the static field applied is B0 , then the electrons precessing around the magnetic field direction produce an induced magnetic field opposing B0 . The total effective magnetic field around the nucleus is then B = B0 − B0 = (1 − σ)B0 , where the parameter σ is called the shielding coefficient. In some cases σ is dependent on the temperature. High-resolution NMR spectroscopy has found that the chemical shifted peaks are also composed of several lines, a result of the spin—spin coupling, which is the second term in the NMR Hamiltonian: X HII = Ii · Jij · Ij . i>j

14.1.3 Organization of the Chapter Section 14.1 thus far has introduced some basic facts of nuclear spins and atomic physics. In Section 14.2, we give background on what quantum computing is about, and introduce universal quantum gates based on liquid NMR.


473

Section 14.3 describes the most recent progress in solid-state NMR quantum gate controls and designs. Sections 14.4 and 14.5 explain applications of the NMR quantum computer to Shor’s algorithm and a lattice-gas algorithm.

14.2 Basic Technology Used in Quantum Computation with NMR 14.2.1 Introduction to Quantum Computation Quantum mechanics is one of the revolutionary scientific discoveries of the 20th century. The field of quantum computation, our emphasis in this chapter, was born when the principles of quantum mechanics were introduced to modern computer science. Quantum computation mainly studies the analysis and construction of quantum algorithms with an eye toward surpassing the classical counterparts. Another tightly connected field is quantum information, which deals more with the storage, compression, encryption, and communication of information by quantum mechanical means [40, 7]. Quantum teleportation [6, 11] and quantum cryptography [5, 29] are two of the most known subjects of this field. Modern computer science emerged when the eminent British mathematician Alan Turing invented the concept of the Turing machine (TM) in 1936 [103]. Although very simple and primitive, the TM captures the essence of computation. It serves as the universal model for all known physical computation devices. For many years, quantum effects had never been considered in the theory of computation, until the early 1980s. Benioff [4] first coined the term of quantum Turing machine (QTM). Motivated by the problem that classical computers cannot simulate quantum systems efficiently, Feynman [35] posed the quantum computer as a solution. Now we know that, in terms of computability, quantum computers and classical computers possess exactly the same computational power. But in terms of computational complexity, which measures the efficiency of computation, there are many exciting examples confirming that quantum computers do solve certain problems faster. The two most significant ones are Shor’s factorization algorithm [96] and Grover’s search algorithm [46], among other examples such as the Deutsch—Jozsa problem [24], the Bernstein—Vazirani problem [9], and Simon’s problem [97]. Current physical realization of quantum computers follows the quantum circuit model [23], instead of the QTM model. The quantum circuit model is another fundamental model of computation, which is equivalent to the QTM model [118], but easier to implement. This model shares many common features of classical computers. In a classical computer, information is encoded

474


H NOT gate

Rθ

Hadamard gate

phase gate

Rθ CNOT gate

controlled-phase gate

Fig. 14.4 Circuit diagrams of the NOT/Hadamard/phase/CNOT/controlled-phase gate.

in multibit binary states (0 or 1), transferred from one register to another, and processed by logic gates in concatenation. In a quantum computer, information is represented by the quantum states of the qubits, and manipulated by various quantum control mechanisms. Those control mechanisms trigger quantum operations to process information in a way resembling the gates in a classical computer. Such quantum operations are called quantum gates and a series of quantum gates in concatenation constitutes a quantum circuit [112]. However, because of the special effects of quantum mechanics, major distinctions exist. In contrast to a classical system, a quantum system can exist in different states at the same time, an interesting phenomenon called superposition. Superposition enables quantum computers to process data in parallel. That is why a quantum computer can solve certain problems faster than a classical computer. From now on, we use the Dirac bra—ket notation. In this notation a pure one-qubit quantum state can be written as |φi = a|0i + b|1i. Here |0i and |1i are the two basis states of the qubit, for example, in NMR, the spin-up and spin-down states, and a, b ∈ C with |a|2 + |b|2 = 1. When we make a measurement of a qubit, the result might be either |0i or |1i, with probabilities |a|2 and |b|2 , respectively. More generally, a string of n qubits P11...1 can in any state of the form |ψi = x=00...0 ψx |xi, where ψx ∈ C and P exist |ψx |2 = 1. When we make a measurement on |ψi, it collapses to |xi, one of the 2n basis states, with probability |ψx |2 . This indeterministic nature makes the design of efficient quantum algorithms highly nontrivial. Another distinctive feature of the quantum circuit is that the operations performed by quantum gates must be unitary (U † U = I). It is the natural consequence of the unobserved quantum systems evolving according to the Schrödinger equation. A quantum gate may operate on any number of qubits. Here are some examples (cf. Figure 14.4 for the circuit diagrams).


475

∙

¸

01 . 10 1 2. The Hadamard gate H: H|0i = √2 (|0i + |1i), H|1i = √12 (|0i − |1i), or 1. NOT gate Λ0 : Λ0 |0i = |1i, Λ0 |1i = |0i, or Λ0 = ∙ ¸ 1 1 1 H=√ . 2 1 −1

3. One-qubit phase gate Rθ : Rθ |0i = |0i, Rθ |1i = eiθ |1i, or ∙ ¸ 1 0 Rθ = . 0 eiθ 4. Two-qubit controlled-NOT (CNOT) gate Λ1 : Λ1 |00i = |00i, Λ1 |01i = |01i, Λ1 |10i = |11i, Λ1 |11i = |10i, or ⎡ ⎤ 1000 ⎢0 1 0 0⎥ ⎥ Λ1 = ⎢ ⎣0 0 0 1⎦. 0010

5. Two-bit controlled-phase gate Λ1 (Rθ ), where Rθ is the one-bit phase gate: Λ1 (Rθ )|00i = |00i, Λ1 (Rθ )|01i = |01i, Λ1 (Rθ )|10i = |10i, Λ1 (Rθ )|11i = eiθ |11i, or ⎤ ⎡ 1 0 0 1 ⎢0 1 0 0⎥ ⎥ Λ1 (Rθ ) = ⎢ ⎣ 0 0 1 0 ⎦. 0 0 0 eiθ The one-qubit and two-qubit quantum gates are of particular importance to the construction of a quantum computer, because of the following universality result.

Theorem 14.2.1. (DiVincenzo [2, 25]) The collection of all the one-qubit gates and the two-qubit CNOT gate suffice to generate any unitary operations on any number of qubits. Figure 14.5 illustrates, as an example, how to generate the two-qubit controlled-phase gate using 2 CNOT gates and 3 one-qubit phase gates. The controlled-phase gate is an important building block for the quantum Fourier transform (cf. Figures 14.14 and 14.15). The standard procedure of executing a quantum algorithm on a quantum circuit usually follows these steps. 1. Initialize the qubits. 2. Apply a proper sequence of quantum gates on the qubits. 3. Measure the qubits. We address the details of these steps in the scenario of NMR technology.

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Rθ

2

Rθ

Rθ

2

R− θ

2

Fig. 14.5 Construction of the controlled-phase gate with CNOT gates and phase gates.

14.2.2 Realization of a Qubit As mentioned in Section 14.1, NMR quantum computing is accomplished by using the spin-up and spin-down states of a spin- 12 nucleus. A molecule with several nuclear spins may work as a quantum computer where each spin constitutes a qubit. In fact, NMR has a long history in information science. Back in the 1950s, nuclear spins were already used for information storage in computers. Liquid NMR receives more interest due to its mature technology and readiness for application. For now, spin- 12 nuclei such as protons and 13 C are preferred because they naturally represent a qubit, but multilevel qubits formed by spin-n nuclei, n = 1, 2, . . . , may provide more freedom in the future. Through careful design, the potential qubits or nuclei are configured with different resonance frequencies and can be distinguished from each other. In a low-viscosity liquid, dipolar coupling between nuclei is averaged away by the random motion of the molecules. The J-coupling (scalar coupling) dominates the spin—spin interaction, which is an indirect through-bond electronic interaction. Previously, a very difficult part of the system operation was to set the quantum system to a special state (or to initialize it). Now a very complicated technology has been developed to solve this problem. Figure 14.6 shows the structure of a trichloroethylene (TCE) molecule and a chloroform molecule used in NMR quantum computers. The hydrogen nucleus (proton) and two 13 C nuclei in a TCE molecule form three qubits which can be manipulated, and the chloroform molecule provides two qubits. The sample used by an NMR quantum computer has a large number (∼1023 ) of such molecules. This is also called a bulk quantum computer. Although most molecules are in a totally random state at room temperature, there are still a small amount of spins standing out and serving our purpose. Theoretically, we use a statistical spin state called a pseudo-pure state, which has the same transformation property as that of a pure quantum state. Let |φi = a|0i + b|1i be the state of a single qubit, |0i for spin-up and |1i for spin-down. We also assume that a is real because only the relative phase is important. Thus this state can be represented using two angles θ and ψ: θ θ |φi = cos |0i + eiψ sin |1i, 2 2

(14.4)


477

Cl

Cl 13

C

H

13

Cl

C

Cl

Cl

13

C

Cl

H

Fig. 14.6 The molecule structure of a candidate 3-qubit quantum system, trichloroethylene (left), and a candidate 2-qubit quantum system, chloroform. The trichloroethylene molecule has two labelled 13 C and a proton, all having one-half-spin nuclei. By considering the P static magnetic fieldPand spin—spin interaction, its Hamiltonian can be written as P H = − 3i=1 gni βni Ii · B + 2i=1 3j=i+1 Ii · Ji,j · Ij . The chloroform has one labelled 13 C and one proton.

where θ ∈ [0, π] and ψ ∈ [0, 2π). If we think of |0i and |1i as the standard basis in C2 , the quantum state corresponds to a unit vector in C2 . For the study of NMR spectroscopy with many nuclei, density matrices are preferred and are often written as the linear combination of product operators [83]: ρ = |φihφ| " cos2 θ = iψ sin2θ e 2

e−iψ sin2 θ sin2

θ 2

#

= I0 + sin θ cos ψIx + sin θ sin ψIy + cos θIz ,

(14.5)

where the product operators are defined as ∙ ¸ ∙ ¸ ∙ ¸ ∙ ¸ 1 10 1 01 1 0 −i 1 1 0 I0 = , Ix = , Iy = , Iz = . (14.6) 2 01 2 10 2 i 0 2 0 −1 They are different from the Pauli matrices only by a constant factor and share the similar commutative law. Upon collecting all the coefficients of Ix , Iy , and Iz together, we obtain a vector v = [sin θ cos ψ

sin θ sin ψ

cos θ]T ,

which is called a Bloch vector. In essence, we have defined a mapping from the set of unit vectors |φi ∈ C2 to the set of unit vectors v ∈ R3 . We have good reasons to ignore the coefficient of I0 , because it has no effect on the spectroscopy and remains unchanged under any unitary transformation. Each Bloch vector determines a point on the unit sphere, called the Bloch sphere, which is displayed in Figure 14.7 [69, 83]. Bloch vectors have proven to be a very good tool for NMR quantum operations.

478


0 Z θ

X

Y ψ

1 Fig. 14.7 The Bloch sphere representation of a quantum state.

The mapping defined above is surjective, because every point on the Bloch sphere gives rise to a unit vector v = [sin θ cos ψ sin θ sin ψ cos θ]T for some pair of (θ, ψ). Conversely, if v(θ0 , ψ 0 ) = v(θ, ψ), we get ⎧ ⎨ cos θ = cos θ0 , sin θ cos ψ = sin θ0 cos ψ 0 , (14.7) ⎩ sin θ sin ψ = sin θ0 cos ψ 0 ,

which can be used to show that the mapping is also injective if we identify all pairs of (0, ψ) with one point and all pairs of (π, ψ) with another point. In fact, these two sets correspond to two states |0i and |1i, respectively.

14.2.3 Transformation of Quantum States: SU(2) and SO(3) When a quantum operation is applied to a quantum system, it may change the quantum state of the system from one to another. The representation of the operation depends on how the quantum state is represented. For example, (14.4) leads to an operator or matrix U which connects the new and old states of a single-spin quantum system: |φ0 i = U |φi, where |φ0 i and |φi are the quantum states after and before the operation, respectively. The fact that both states are unit vectors implies that U is


479

a 2 × 2 unimodular complex matrix. Moreover, U is also unitary; that is, U ∈ SU(2),2 a Lie group endowed with a certain topology. If the quantum state is represented by a three-dimensional Bloch vector, the effect of a unitary operation can be viewed as that of a rotation which rotates the Bloch sphere, and the operator is represented by a 3 × 3 real matrix S. If the quantum system has states v and v0 in Bloch vector form before and after the operation, respectively, then v0 = Sv. The matrix S is a proper rotation matrix; that is, S ∈ SO(3).3 It is isometric and preserves the threefold product. If both S and U represent the same physical operation, such as a transformation induced by a series of pulses in NMR, there must be a connection between them. One can show that there is a mapping R from SU(2) to SO(3) such that S = R(U ), for any U ∈ SU(2) and its corresponding Bloch-sphere representation S [88]. Simple computation shows that the entry of matrix S = R(U ) at the kth row and ith column is given as Ski = Tr(σk U Ii U † ),

(14.8)

where σk are the Pauli matrices,4 and Tr is the trace operator. It can also be shown that R is a two-to-one homomorphism between SU(2) and SO(3) with kernel ker(R) = {I, −I}. It coincides with the fact that U and −U in SU(2) represent the same operation because only the relative phase matters. This mapping is also surjective, so it defines an isomorphism from the quotient group SU(2)/ker(R) to SO(3). We provide a more detailed discussion about this isomorphism in the appendix. It is known that any U ∈ SU(2) can be written into an exponential form parameterized by an angle θ ∈ [0, 2π) and a unit vector n such that U (θ, n) = e−i(θ/2)n·σ " cos θ2 − in3 sin θ2 = sin θ2 (n2 − in1 )

− sin θ2 (n2 + in1 ) cos θ2 + in3 sin θ2

#

(14.9)

= cos θ2 I − i sin θ2 n · σ,

where σ = [σx , σy , σz ]. With this parameterization of SU(2), entries of S = R(U ) can be computed using (14.8) as 2 SU(n) is the special unitary group of n × n matrices. An n × n matrix A ∈ SU(n) if and only if A is unitary (i.e., A · A† = In , where A† is the Hermitian adjoint of A, and det A = 1). 3 SO(3) denotes the special orthogonal group of 3×3 matrices. An n×n matrix A ∈ SO(n) if and only if A is real, AAT = In , and det A = 1. ¤ ¤ ¤ £ £ £ 4 The Pauli matrices are σ = 0 1 , σ = 0 −i , and σ = 1 0 . x y z 0 −1 10 i 0

480


Sij = R(U )ij = cos θ δij + (1 − cos θ)ni nj +

3 X

sin θ

ikj nk .

(14.10)

k=1

It should be noted now that S coincides with a rotation about the axis along n with an angle θ in the three-dimensional Euclidean space after comparing Sij with the standard formula of a rotation matrix. This interpretation is important in understanding the terminologies used in NMR. For example, the rotations around x-, y-, and z-axes (x/y/z-rotations) with an arbitrary angle θ define the following three unitary operators in SU(2), respectively. " # −i sin θ2 cos θ2 −iθσx /2 Xθ = e = , (14.11) −i sin θ2 cos θ2 # " cos θ2 − sin θ2 −iθσy /2 Yθ = e , (14.12) = sin θ2 cos θ2 " # e−iθ/2 0 −iθσz /2 Zθ = e = . (14.13) 0 eiθ/2

14.2.4 Construction of Quantum Gates From Theorem 14.2.1, we know that the collection of all the one-qubit gates and the two-qubit CNOT gate are universal. In addition, the following fact [85, p. 175] holds for one-qubit quantum gates. Theorem 14.2.2. Suppose U is a unitary operation on a single qubit. Then there exist real numbers α, β, γ, and δ such that U = eiα Zβ Yγ Zδ . For example, the Hadamard gate H can be decomposed as H = eiπ/2 Yπ/2 Zπ . Clearly, the x/y/z-rotation gates provide building blocks sufficient to construct any one-qubit unitary gate. In this section, we show how to realize these one-qubit rotation gates and the two-qubit CNOT gate using NMR. We also show how to decouple the interaction between two spins, a process called refocusing [85].

14.2.4.1 One-Qubit Gates A single-spin system has Hamiltonian H = −μ · B, where μ is the magnetic moment, and B = B0 ez + B1 (ex cos(ωt) + ey sin(ωt)) (14.14)


481

is the magnetic field applied. B0 , a large constant, is the amplitude of the static magnetic field, and B1 is the amplitude of the oscillating magnetic field in the x—y plane. When B1 = 0, the Hamiltonian and Schrödinger equations can be obtained as ([85]) ω0 H= (14.15) σz 2 and (14.16) i∂t |ψ(t)i = H|ψ(t)i, respectively, where ~ has been divided from both sides in the second equation and we take ~ away from H in the first one just for simplicity. The Larmor frequency ω0 = −B0 γ is defined by the nuclei and the magnetic field; see (14.3). Assume that the initial state is |ψ0 i = a0 |0i+b0 |1i. Then the evolution of the quantum state of the spin and the density matrix can be solved directly and given as |ψ(t)i = e−iω0 σz t/2 |ψ0 i ∙ −iω t/2 ¸∙ ¸ e 0 a0 0 = b0 0 eiω0 t/2 ∙ ¸ 1 0 = e−iω0 t/2 |ψ0 i, 0 eiω0 t ρ(t) = e−itH ρ(0)eitH .

This evolution is also called a chemical shift evolution, resembling the precessing of a magnet in a static field. Recall the Bloch vector on the Bloch sphere. It is exactly Zθ , the rotation operator around the z-axis with θ = ω0 t. To achieve an x-rotation operator, we need a small magnetic field transverse to the z-direction to control the evolution of the quantum state. The Hamiltonian is given as in (14.14) by choosing B1 different from zero: H = −μ · B =

ω0 ω1 σz + (σx cos(ωt) + σy sin(ωt)) , 2 2

where ω1 depends on the x—y plane component B1 of the magnetic field, ω1 = −B1 γ. To solve the Schrödinger equation, we put |ψ(t)i in a “frame” rotating with the magnetic field around the z-axis at frequency ω, |φ(t)i = eiωtσz /2 |ψ(t)i. With this substitution, the Schrödinger equation (14.16) becomes ω i∂t |φ(t)i = (eiωσz t/2 He−iωσz t/2 − σz )|φ(t)i. (14.17) 2 Using properties eiωσz t/2 σz e−iωσz t/2 = σz , eiωσz t/2 σx e−iωσz t/2 = σx cos(ωt) − σy sin(ωt),

eiωσz t/2 σy e−iωσz t/2 = σx sin(ωt) + σy cos(ωt),

(14.18)

482


we obtain i∂t |φ(t)i =

µ

¶ ω0 − ω ω1 σz + σx |φ(t)i, 2 2

|φ(t)i = e−i((ω0 −ω)σz /2+ω1 σx /2)t |φ(0)i. We know from (14.9) that this is a rotation around the axis µ ¶ 1 ω1 n= q z+ x . ω0 − ω 1 1 + ( ω0ω−ω )2

(14.19)

(14.20)

An important case is ω0 = ω, also called the resonance case where its name comes from the zero denominator in (14.20). By (14.19), we see that a relatively weak transverse magnetic field causes a rotation around the x-axis:

|ψ(t)i = e−iω0 σz t/2 |φ(t)i = e−iω0 tσz /2 e−iω1 tσx /2 |φ(0)i = Zθ Xβ |ψ(0)i, (14.21) where Xβ = e−iω1 tσx /2 , β = ω1 t. By applying another Z−θ , we obtain a rotation Xβ as desired. Because the frequency of the precession is in the radio frequency band, the field applied is called an RF pulse. When |ω0 − ω| À ω1 , the rotation axis direction is almost along z and the RF pulse has no effect on it: |ψ(t)i = e−iωσz t/2 |φ(t)i ≈ e−iω0 tσz /2 |ψ(0)i = Zω0 t |ψ(0)i, thus we can tell one qubit from another because their resonance frequencies are designed to be different. There are still cases where the difference of resonance frequencies between spins is not large enough. The RF pulse may cause similar rotations on all those spins. To avoid or at least minimize it, a soft pulse is applied instead of the so-called hard pulse. It is a pulse with a longer time span and weaker magnetic field, in other words, a smaller ω1 . This strategy makes these “close” qubits fall into the |ω0 − ω| À ω1 case. If we change the magnetic field to B = B0 ez + B1 (ex cos(ω0 t + α) + ey sin(ω0 t + α)),

(14.22)

the Hamiltonian will become H=

ω0 ω1 σz + (σx cos(ω0 t + α) + σy sin(ω0 + α)), 2 2

(14.23)

where ω1 is defined as before. The RF field is almost the same as (14.14) in the resonance case except for a phase shift. Using the same rotation frame as before with ω = ω0 , we obtain i∂t |φ(t)i =

ω1 (σx cos(α) + σy sin(α))|φ(t)i, 2

(14.24)


483

after simplification. After time duration t, the new system state is given as |φ(t)i = e−i(ω1 /2)(σx cos(α)+σy sin(α))t |φ(0)i,

(14.25)

and the evolution operator can be computed using (14.9) as Uθ/2,α = e"−i(ω1 /2)(σx cos(α)+σy sin(α))t # cos( θ2 ) −i sin( θ2 )e−iα = , −i sin( θ2 )eiα cos( θ2 )

(14.26)

where θ = ω1 t. This is a one-qubit rotation operator, and is sometimes called a Rabi rotation gate. When α = π/2, # " cos( θ2 ) − sin( θ2 ) Uθ/2,π/2 = sin( θ2 ) cos( θ2 ) (14.27) = Yθ . We have achieved a y-rotation operator just by adding a phase shift to the RF field.

14.2.4.2 Two-Qubit Gates The construction of a two-qubit gate requires the coupling of two spins. In a liquid sample of NMR, J-coupling is the dominating coupling between spins. Under the assumption that the resonance frequency difference between the coupled spins is much larger than the strength of the coupling (a so-called weak coupling regime), the total Hamiltonian of a two-spin system without transverse field may be given as H=

1 1 1 ω1 σz1 + ω2 σz2 + Jσz1 σz2 , 2 2 2

(14.28)

where ωi is the frequency corresponding to spin i, σzi is the z-projection operator of spin i, for i = 1, 2, and J is the coupling coefficient. Take the chloroform in Figure 14.6, for example [16, 69]. In an 11.7 T magnetic field, the precession frequency of 13 C is about 2π × 500 MHz and the precession frequency of the proton is about 2π × 125 MHz. The coupling constant J is about 2π × 100 Hz. Here we set B1 = 0, which means no transverse magnetic field is applied and those terms such as σx , σy do not appear. The remaining terms in the Hamiltonian only contain operators σz1 or σz2 , which are commutative. Thus, we can obtain the eigenstates and eigenvalues of this two-spin system and we map the set of eigenstates to the standard basis of C4 , as follows.

484


⎡ ⎤ 1 ⎢0⎥ ⎥ |00i = ⎢ ⎣0⎦, 0

⎡ ⎤ 0 ⎢1⎥ ⎥ |01i = ⎢ ⎣0⎦, 0

⎡ ⎤ 0 ⎢0⎥ ⎥ |10i = ⎢ ⎣1⎦, 0

⎡ ⎤ 0 ⎢0⎥ ⎥ |00i = ⎢ ⎣0⎦; 1

(14.29)

H|00i = k00 |00i, k00 = 12 ω1 + 12 ω2 + 12 J; H|01i = k01 |01i, k01 = 12 ω1 − 12 ω2 − 12 J;

H|10i = k10 |10i, k10 = − 12 ω1 + 12 ω2 − 12 J;

(14.30)

H|11i = k11 |11i, k11 = − 12 ω1 − 12 ω2 + 12 J.

Because the matrix is diagonal, the evolution of this two-spin system can be easily derived as ⎤ ⎡ −ik t e 00 ⎥ ⎢ e−ik01 t ⎥ |ψ(0)i. (14.31) |ψ(t)i = e−iHt |ψ(0)i = ⎢ −ik10 t ⎦ ⎣ e −ik11 t e We can also rewrite the one-qubit rotation operators for this two- 12 -spin system in matrix form with respect to the same basis: ⎤ ⎡ −iπ/4 e ⎥ ⎢ e−iπ/4 1 ⎥, (14.32) =⎢ Zπ/2 iπ/4 ⎦ ⎣ e iπ/4 e ⎡

⎢ 2 =⎢ Z−π/2 ⎣

⎤

eiπ/4

e−iπ/4 eiπ/4 ⎡

e−iπ/4 ⎤

1 −1 √ ⎥ ⎢ 2 1 2 ⎥, ⎢ 1 = Yπ/2 1 −1 ⎦ 2 ⎣ 1 1 ⎡ ⎤ 1 1 √ ⎥ 2⎢ 2 ⎢ −1 1 ⎥, Y−π/2 = ⎣ 1 1⎦ 2 −1 1

⎥ ⎥, ⎦

(14.33)

(14.34)

(14.35)

where Zθi is the rotation operator for spin i with angle θ around the z-axis while keeping another spin unchanged, and all Yθi are similarly defined operators about the y-axis; see (14.12). A careful reader may raise issues about the one-qubit gate we have obtained in Section 14.2.4.1 because the coupling


485

between two qubits always exists and has not been considered. We need to turn off the coupling when we only want to operate one spin but the coupling is nonnegligible. This is in fact one of the major characteristic difficulties associated with the NMR quantum computing technology. A special technology called refocusing is useful. It works as follows. We apply a soft π pulse on the spare spin that we don’t want to change at the middle point of the operation time duration while we are working on the target spin. The effect is that the coupling before the pulse cancels the one after the pulse, so the result of no-coupling is achieved. Another π pulse will be needed to turn the spin back. All pulses are soft. This technology is so important that we now state it here as a theorem. Theorem 14.2.3. Let H = (ω1 /2) σz1 + (J/2) σz1 σz2 + A be a given Hamiltonian, where A is a Hamiltonian that does not act on spin 1 and commutes with σz2 . Then the evolution operators of A and H satisfy e−iAt = −Xπ1 e−iHt/2 Xπ1 e−iHt/2 ;

(14.36)

that is, the collective evolution of the quantum system with Hamiltonian H and additional two Xπ1 -pulses at the middle and the end of the time duration, equals that of a system with Hamiltonian A (up to a global phase shift π, or a factor −1). Proof. Assume that the time duration is t and denote U for U = Xπ1 e−iHt/2 Xπ1 e−iHt/2 .

(14.37)

1

Note that Xπ1 = e−i(π/2)σx and it commutes with A which contains no operators acting on spin 1, thus 1

1

2

1

1

2

1

2

U = Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 e−iAt . (14.38) It suffices to prove that the part before e−iAt satisfies 1

1

2

1

B = Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 = −I. (14.39) We first check the effect of B on the four basis vectors. We have 1

1

2

1

1

2

B|11i = Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 |11i 1

1

2

= e−i((−ω1 +J)/4)t (−i)Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 |01i = (−i)e−i((−ω1 +J)/4)t Xπ1 e−i((ω1 −J)/4)t |01i

= (−i)2 |11i = −|11i,

(14.40)

486

Z. Zhang, G. Chen, Z. Diao, P.R. Hemmer 1

1

2

1

1

2

B|01i = Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 |01i 1

1

2

= e−i((ω1 −J)/4)t (−i)Xπ1 e−i((ω1 /2)σz +(J/2)σz σz )t/2 |11i = (−i)e−i((−ω1 +J)/4)t Xπ1 e−i((−ω1 +J)/4)t |11i

= (−i)2 |01i = −|01i,

(14.41)

and similarly, B|10i = −|10i,

(14.42)

B|00i = −|00i.

In the computation above, we have used the fact that Xπ1 has no effect on the second spin and the four basis vectors |00i, |01i, |10i and |11i are the eigenstates of the operator (ω1 /2) σz1 + (J/2) σz2 σz1 . The result shows that B = −I, and we are done. t u When the Hamiltonian is given in the form as (14.28), the above theorem tells us that both the chemical shift evolution (precession) and the J-coupling effect on spin 1 are removed and only the term (ω2 /2) σz2 remains. We obtain a z-rotation of spin 2 while freezing spin 1. By combining it with several hard pulses, we can also achieve any arbitrary rotation on spin 2 with the motion of spin 1 frozen [73]. Similar computation shows that a hard π pulse applied at the middle point of the time duration cancels the chemical shift evolution of both spins. This can be seen by checking the identity ⎡

⎢ e−iHt/2 Xπ1 Xπ2 e−iHt/2 = ⎢ ⎣

e−iJt/2 eiJt/2 iJt/2

e e−iJt/2

⎤

⎥ ⎥. ⎦

(14.43)

Another hard π pulse can rotate two spins back, so we have achieved an evolution which has only the J-coupling effect, denoted by Zθ : ⎤ ⎡ e−iθ/2 ⎥ ⎢ eiθ/2 ⎥, Zθ = ⎢ ⎦ ⎣ iθ/2 e −iθ/2 e and when θ = π/2,

⎡

⎢ Zπ/2 = ⎢ ⎣

⎤

e−iπ/4 eiπ/4 iπ/4

e

e−iπ/4

⎥ ⎥. ⎦

(14.44)


487

Zπ/2 Zπ/2 Yπ/2

Z−π/2

Y−π/2

Fig. 14.8 The quantum circuit used to realize a quantum controlled-not gate.

Although we give only an example of the 2-qubit system in the above, the reader should note that a general method is available to reserve only the couplings wanted while keeping all the others cancelled for multiqubit systems [57, 71, 73]. Combining operators in (14.32) through (14.35) and (14.44), we can now construct a CNOT gate as in Figure 14.8 which includes four 1-qubit π/2 rotations around y- or z-axes and one 2-qubit π/2 rotation. The total operator, denoted by CN , can be computed as ⎡ ⎤ 1 ⎥ π ⎢ 1 1 2 2 2 ⎥, CN = Zπ/2 (14.45) Y−π/2 Z−π/2 Zπ/2 Yπ/2 = e− 4 i ⎢ ⎣ 0 1⎦ 10

which is a CNOT gate up to a phase of −π/4 [69]. We have shown how to construct one-qubit gates and the two-qubit CNOT gate using the NMR technology. The simple pulse design works fine in ideal situations. In practice, errors arise from various factors. Decoherence causes the lost of quantum information with time. Thus, all operations should be completed within a short time, roughly constrained by the energy relaxation time T1 and the phase randomization time T2 . Again, take the chloroform for an example. For protons, T1 ≈ 7 sec and T2 ≈ 2 sec; for carbons, T1 ≈ 16 sec and T2 ≈ 0.2 sec [16, 69]. The pulses have to be short enough so that all the pulses can be jammed in the time window. Ideally, a pulse can be completed quite fast, but this may incur undesirable rotations in other qubits because the frequency bandwidth is inversely proportional to the time length of the pulse. A shorter and stronger pulse will have a wider frequency band that may cover the resonance frequency of another spin, called cross-talking. It should also be noted that both T1 and T2 are defined and measured in a simplified situation, and they can only be used as an approximation of the decoherence rate for the quantum computation. Coupling is also a problem that makes the pulse design much more complicated. Finally, any experimental facility is not

488


perfect, which may introduce more errors. Typical error resources include inhomogeneities in the static and RF field, pulse length calibration errors, frequency offsets, and pulse timing/phase imperfections. If the quantum circuit can be simplified and the number of gates needed is reduced, the requirements on the pulses can be alleviated. Mathematicians are looking for methods to find time-optimal pulse sequences [43, 60, 61, 99], with the goal of finding the shortest path between the identity and a point in the space of SU(n) allowed by the system and the control Hamiltonians. In addition, NMR spectroscopists have already developed advanced pulse techniques to deal with system errors such as cross-talking and coupling. They turn out to work well and are now widely used in NMR quantum computation. Such techniques include composite pulses [21, 36, 72, 54, 55, 108] and pulse shaping. The latter consists mainly of two methods: phase profiles [89] and amplitude profiles [39, 66].

14.2.5 Initialization An NMR sample eventually will go into its equilibrium state when no RF pulse is applied for a long time. Then the density matrix is proportional to e−H/kT , according to the Boltzmann distribution, where k = 1.381 × 10−23 J/K and T is the absolute temperature. Normally, the environment temperature is far larger than the energy difference between the up and down states of the spin, and H/kT is very small, about 10−4 . We also make the assumption that the coupling terms are small enough compared with the resonant frequency, thus we can make a reasonable approximation of the equilibrium state density matrix of a system with n spins: ρeq =

1 e−H/kT ≈I− ( 1 σz1 + kT tr(e−H/kT )

2 2 σz

+ ··· +

n n σz ).

(14.46)

In the four operators appearing in the density matrix (14.5), only those with zero traces can be observed in NMR. The operator I0 is invisible, and moreover, it remains invariant under any unitary similarity transformation. Therefore, we only need to take care of the zero-trace part of the initial density matrix, noting that only that part (called the deviation) is effective. Most algorithms prefer an initial state such as ρ0 =

1− I + |00 · · · 0ih0 · · · 00|, 2n

which is an example of the so-called pseudo-pure states, corresponding to the pure state |00 · · · 0i. To initialize the system to a pseudo-pure state as above, we may use a scheme called averaging. Let us explain this for a 2-spin system. Suppose we


have three 2-spin subsystems with density matrices ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a000 a000 a000 ⎢0 b 0 0⎥ ⎢0 c 0 0⎥ ⎢0 d 0 0⎥ ⎥ ⎥ ⎥ ρ1 = ⎢ ρ2 = ⎢ ρ3 = ⎢ ⎣0 0 c 0⎦, ⎣0 0 d 0⎦, ⎣0 0 b 0⎦, 000d 000b 000c

489

(14.47)

respectively, where a, b, c, and d are nonnegative, and a + b + c + d = 1. These are three diagonal matrices with three of their diagonal elements in cyclic permutation. Now, we mix these three subsystems together (for an n-qubit system, we may have 2n − 1 subsystems) and assume that the three subsystems have the same signal scale. Because the readout is linear with respect to the initial state, we are in fact working on a system with an effective initial density matrix ⎤ ⎡ 3a 3 ⎥ 1⎢ 1X b+c+d ⎥ ρi = ⎢ ⎦ ⎣ b+c+d 3 i=1 3 b+c+d ⎡ ⎤ 4a − 1 0 0 0 b+c+d 1⎢ 0 0 0 0 ⎥ ⎥, = I+ ⎢ (14.48) ⎣ 0 0 0 0 ⎦ 3 3 0 0 0 0

which is a pseudo-pure state corresponding to |00 · · · 0i. Various methods have been developed to achieve this effect of averaging. Because ρ1 , ρ2 , and ρ3 differ only by a permutation of the diagonal elements, a sequence of CNOT pulses can be used to transform one to another. In most cases, we only have one sample; the same algorithm can be repeated on that sample three times but with different initial states ρ1 , ρ2 , and ρ3 , respectively. At last, after all three outputs are obtained and added together (average), we achieve the same result as what we will get when the algorithm is employed on a system with the expected initial state |00 · · · 0i. This is called “temporal averaging” [63]. Gradient fields can also be used to divide the sample into different slices in space which are prepared in different initial states, and the averaging is realized spatially, called “spatial averaging” [19]. The number of the experiments and pulses needed grows very large when the number of qubits increases. For example, nine experiments are combined in order to prepare one pseudo-pure state for a 5-qubit system and 48 pulses are used to form one pseudo-pure state in a 7-qubit system [41] after modifications such as logical labeling [42, 106] and selective saturation [62].

490


14.2.6 Measurement An NMR computer differs from other quantum computers in that it works on an ensemble of spins instead of just a single one. It produces an observable macroscopic signal that can be picked up by a set of coils positioned on the x—y plane, as shown in Figure 14.3. The signal measures the change rate of the magnetic field created by a large number of spins in the sample rotating around the z-axis, called free induction decay (FID). Due to relaxation, peaks of the Fourier transform of the signal, or spectra, have width. However, we do not need to worry about that because it does not make any substantial difference in our discussion here. One disadvantage is that the readout from NMR is an average of all the possible states, in contrast to most existing quantum algorithms that ask for the occurrence of only a single state. But it is possible for one to modify ordinary quantum algorithms to make NMR results usable. The magnetization detected by the coil in Figure 14.3 is proportional to the trace of the product of the density matrix with σ+ = σx + iσy : Mx + iMy = nV hμx + iμy i = nV γ~ Tr(ρ(σx + iσy )),

(14.49)

where γ is the magnetogyric ratio as in (14.3) and ρ is the density matrix. When the external RF magnetic field is removed, the density matrix will change according to the system’s Hamiltonian as we discussed earlier. If we decompose the density matrix into a sum of product operators as in (14.5), only Ix and Iy contribute to the readout. We cannot “see” the coefficients of I0 and Iz . Recall (14.18): if a one-spin system begins from density matrix ρ0 = I0 + sin θ cos ψIx + sin θ sin ψIy + cos θIz , the magnetization will rotate with the resonant frequency as Mz + iMy = C Tr(e−iHt ρ0 eiHt σ+ ) = C Tr(e−iHt (I0 + sin(θ) cos(ψ)Ix + sin(θ) sin(ψ)Iy + cos(θ)Iz )eiHt σ+ ) = C Tr((sin θ cos ψ(cos(ωt)Ix + sin(ωt)Iy ) + sin θ sin ψ(cos(ωt)Iy − sin(ωt)Ix ))σ+ )

(14.50)

= C sin θ ei(ωt+ψ) ,

where C = nV γ~. This rotating magnetization will introduce an oscillating electric potential in the receiver coils, which will be processed by a computer to generate the spectra. Note that the signal is proportional to sin θ. If an x rotation with angle π/2 is applied on the spin before the measurement, the magnetization will become √ 2 Mz + iMy = C(sin θ − i cos θ)eiωt . 2


491

For simplicity, we have chosen ψ = 0. The imaginary part is proportional to the population difference: cos θ = cos2

θ θ − sin2 . 2 2

Computation of a two-spin system is complicated, so we only give some partial results here. The purpose is to point out what methodology is used. We still use the basis given by (14.29) and the Hamiltonian in (14.28). The system begins from a density matrix as ⎤ ⎡ ρ11 ρ12 ρ13 ρ14 ⎢ ρ21 ρ22 ρ23 ρ24 ⎥ ⎥ (14.51) ρ0 = ⎢ ⎣ ρ31 ρ32 ρ33 ρ34 ⎦ . ρ41 ρ42 ρ43 ρ44

The operator σ+ is a summation of operators from the two subsystems: 1 2 σ+ = σ ⎡+ + σ+ ⎤ 0220 ⎢0 0 0 2⎥ ⎥ =⎢ ⎣0 0 0 2⎦. 0000

(14.52)

The magnetization in the x—y plane is composed of four frequencies: Mx + iMy = C Tr(e−iHt ρ0 eiHt σ+ ) = C (ρ31 ei(ω1 +J)t + ρ42 ei(ω1 −J)t + ρ43 ei(ω2 −J)t + ρ21 ei(ω2 +J)t ). (14.53) The spectrum has two pairs of peaks, one pair around the precession frequency ω1 and another pair around ω2 . See Figure 14.9. The splitting is a result of coupling. If the system has more than two spins, the coupling will split a peak into up to 2n−1 peaks where n is the number of spins. We also combine all the constants in C to make the formula concise. Only four of the elements out of the density matrix appear in this spectrum, so we need to design certain control pulses to move the expected information to these four positions where numbers can be shown via free induction signal. If multitests are allowed, theoretically, all the elements of the density matrix can be retrieved [15, 14]. It is also possible to transport the desired information (computational results) to the four positions where the observer can see. A typical pulse used in reading out is a hard Xπ/2 pulse which rotates all the spins about the x-axis with angle π/2. Let us still use two-spin systems as an example. The operation is the tensor product of two x-rotation operators; 1 2 that is, Xπ/2 = Xπ/2 Xπ/2 . The imaginary parts of the four effective elements 0 of the density matrix ρ after the operation, utilizing the fact that the density

492


J

J

ω1

J

J

ω2

Fig. 14.9 Simplified stick spectra of a two-qubit molecule. The two dotted lines show two peaks at ω1 and ω2 , respectively, when no coupling is applied (J = 0). After coupling, every peak is split into two small peaks with the intensities reduced to half.

matrix is Hermitian, are Im(ρ031 ) = 14 (ρ33 + ρ44 − ρ11 − ρ22 − 2 Im(ρ21 ) − 2 Im(ρ34 )),

Im(ρ042 ) = 14 (ρ33 + ρ44 − ρ11 − ρ22 + 2 Im(ρ21 ) + 2 Im(ρ34 )), Im(ρ043 ) = 14 (ρ22 + ρ44 − ρ11 − ρ33 + 2 Im(ρ31 ) + 2 Im(ρ24 )),

(14.54)

Im(ρ021 ) = 14 (ρ22 + ρ44 − ρ11 − ρ33 − 2 Im(ρ31 ) − 2 Im(ρ24 )). Find the sum of Im(ρ031 ) and Im(ρ042 ) and that of Im(ρ043 ) and Im(ρ021 ): Im(ρ031 + ρ042 ) = − 12 (ρ11 + ρ22 − ρ33 − ρ44 ),

Im(ρ043 + ρ021 ) = − 12 (ρ11 − ρ22 + ρ33 − ρ44 ).

(14.55)

Because what the coils pick up is the change rate of the magnetic field rather than the magnetic field itself, the imaginary parts we have listed above are reflected in the real part of the spectra. The computation above shows that the sum of the real parts of each pair of peaks in the spectra is proportional to the population difference between the spin-up and the spin-down states of the corresponding spin.

14.3 Solid-State NMR Liquid NMR, discussed in Section 14.2, has several constraints that make a liquid NMR quantum computer not scalable. At first, as the result of the pseudo-pure state preparation, the signal—noise ratio decreases exponentially when the number of qubits increases, limiting its ability to realize more qubits. Another difficulty arises when we want to control the system as accurately as desired. Because the range of the chemical shift is limited by nature, the number of qubits represented by the same type of nuclei, such as carbon, is constrained as the resonance frequency gaps between any two qubits must be large enough so that we can distinguish the qubits easily and


493

control them with great precision. It is estimated that a quantum computer realized by liquid-state NMR can have at most 10 to 20 qubits. Solid-state NMR has the potential to overcome many of the problems of its liquid-state counterpart as in the preceding paragraph. These advantages are derived partly from the lack of motion of the molecules and partly from the ability to cool to low temperatures. As with many potential solutions, there are tradeoffs to consider. Here we summarize. 1. At low temperatures, near or below that of liquid helium, it is possible to initialize electron spins using the thermal Boltzman distribution. Nuclear spins do not become significantly oriented until much lower temperatures because of their ∼1000 times lower energies, but there are existing pulse RF sequences that can transfer an electron spin orientation to nearby nuclear spins using their mutual spin—spin interaction. In principle, this solves the problem of qubit initialization. In practice, the thermal initialization process can be slow because it depends on the electron spin population lifetime. It is possible to find systems with short electron spin lifetimes, but this will tend to result in faster decoherence of the nuclear spins, because they must be coupled to the electron in order to initialize in the first place. 2. Because the molecules in a solid are usually not tumbling, the dipole coupling between nearby spins does not average out. This has the advantage of making multiqubit gates faster, because the dipole coupling is much larger than the scalar coupling. The orientation-dependent chemical shifts also do not average out, in principle making individual qubits easier to address so that more qubits can be used. Here, it should be noted that custom molecules containing electron spins [111] can be used to enhance this effect. There is a tradeoff to consider, in that the faster interaction with nearby spins provided by dipole coupling can also lead to faster decoherence times. 3. Spin lifetimes in solids can be much longer than in liquids. Lack of molecular motion eliminates the spatial diffusion of spins which is a problem in liquid NMR for times in the range of milliseconds or longer [34]. Phonons can cause decoherence in solids at room temperature, but this can be strongly suppressed at temperatures achievable in liquid helium. It is not unusual to see spin population lifetimes of minutes in solids, especially at low temperatures. Unfortunately, spin coherence times are usually somewhat shorter due to dephasing caused by mutual spin flips through the strong dipole coupling. To eliminate this decoherence mechanism, there are two main approaches. One is to disperse the active molecule, as a dopant in a spin-free host. Actually the host does not need to be completely spin-free provided its spins are far enough off resonance with those of the active molecule. Another technique is to use stoichiometric materials consisting of relatively large unit cells containing many spin-free atoms. The idea for both these approaches is to keep the active nuclei relatively far apart, except for nearest neighbors. In addition to the above differences, nuclei with nonzero spin in solid state can also be used for quantum computation [67] and manipulated similarly to the liquid-state NMR. Because all the nuclei are fixed in space, a static

494


magnetic field with a strong gradient in one direction separates the nuclei into different layers along the direction. Every layer of nuclei can be regarded as a qubit and the qubits have different resonance frequencies as the magnetic field is different from one layer to another. Readouts also can be made to take advantage of the bulk quantum computer much as with the liquid NMR. The signal is picked up using methods such as magnetic resonance force microscopy. There are two types of methods to make such a nuclei arrangement. Crystal, such as cerium-monophosphide (CeP), is a natural choice, where the 1/2 spin 31 P nuclei form periodical layers in the crystal with interlayer distance about 12 ˚ A [45, 116]. Another method is to grow a chain of 29 Si that has 1/2 spin along the static field direction on a base of pure 28 Si or 30 Si which are both 0 spin nuclei [1, 68]. The last one combines the mature crystal growth and processing technology for silicon from the semiconductor industry. Liquid crystal [117] or solid-state samples [70] are also candidates for realizing an NMR quantum computer. Recently, there has been considerable progress made in the area of optically addressed spins in solids. As a result some highly scalable designs have recently come forward that have the potential to eliminate all of the limitations of NMR. Aside from potentially solving NMR’s problems, optical addressing has the important advantage that it would provide an interface between spin qubits and optical qubits, which is essential to interface with existing quantum communication systems, and for quantum networking in general. Optically addressed spins are better known in the literature as spectral hole burning (SHB) materials [81]. Most of these are dopant—host systems that exhibit strong zero-phonon optical absorption lines at low temperature. Due to the inherent inhomogeneity of dopant—host systems it is often found that this optical zero-phonon linewidth is much larger than that of the individual atoms. Furthermore, when these transitions are excited with a narrowband laser, the resonant atoms can be optically pumped into a different ground state, making the material more transparent at the laser frequency. This is known as burning a spectral hole, hence the name SHB. In many SHB materials, the optical pumping is into different ground-state spin sublevels, and hence the hole burning process can initialize spin qubits, as illustrated in Figure 14.10. This type of spin qubit initialization can be much faster than Boltzmann initialization, especially in spin systems with long spin population lifetimes, because the tradeoff between spin lifetime and initialization speed is removed. In addition to optical initialization of spins, the hole burning process can also be used to read out the spin state of the qubits. This happens when the quantum algorithm returns some of the spin qubits to the state that was initially emptied by the laser, resulting in a temporary increase in laser absorption and/or fluorescence that is proportional to the final population of this spin state. Of course, the readout process also reinitializes, so one must


495

Fig. 14.10 (a) The signature of spectral hole burning is a narrowband dip in the optical absorption spectrum. (b) This dip occurs when an optical laser bleaches out an ensemble of atoms at a particular transition frequency. (c) In the case when bleaching is due to spin sublevel optical pumping, it can be used to initialize qubits.

take care to work with a large enough ensemble to achieve the desired readout fidelity. In general, optical readout is orders of magnitude more sensitive than the typical NMR coil, and so it is possible to work with small ensembles consisting of very dilute dopant—host systems that can have very long spin coherence lifetimes. Spin qubit coherence lifetimes in SHB materials can be lengthened by a variety of techniques. In dilute dopant—host systems, the choice of a spin-free or low-spin host has the largest benefit. Examples include praseodymium [50] or europium [32] doped in an yttrium-silicate host (Pr:YSO or Eu:YSO) and nitrogen-vacancy [110] (NV) color centers doped in diamond; see Figure 14.11. In Pr:YSO only the yttrium host nuclei have spin but the magnetic moment is very weak. In NV diamond, the only host spins are ∼1% abundant 13 C which can be virtually eliminated with isotopically pure material. In dopant— host systems dephasing due to host spins is reduced by the so-called frozen core effect [101], wherein the magnetic field generated by the active (qubit) spin system tunes nearby host nuclei out of resonance with the rest of the crystal up to a distance that defines the frozen core radius. This suppresses the energy-conserving mutual spin flips that are the main source of spin decoherence. In Pr:YSO the spin Hamiltonian is given by [77]: ³ ³ ← ³ ← ← →´ → ← →´ → ← → ´ H = B · gJ2 μ2B Λ · B + B · γN E + 2AJ gJ μB Λ · I + I · A2J Λ + T Q · I, (14.56) ← → where the tensor Λ is given by

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Fig. 14.11 (a) Spin sublevels of nitrogen-vacancy (NV) color center in diamond. (b) Spin sublevels of Pr : YSO.

Λαβ =

2J+1 X n=1

h0|Jα |nihn|Jβ |0i ∆En,0

(14.57)

← → In (14.56), E is the 3 × 3 identity matrix, B is the magnetic field, I is the nuclear spin vector, gJ is the Lande g, γN is the nuclear magnetogyric ratio, ← → and AJ is the hyperfine interaction. The term I · T Q · I describes the nuclear ← → electric quadrupole interaction and A2J I · Λ · I is the second-order magnetic hyperfine or pseudoquadrupole interaction. Recently, a spin coherence lifetime of 1/2 minute has been observed in Pr:YSO [37]. This impressive result is made possible by combining two techniques. The first technique involves magnetically tuning the qubit spin to a level anticrossing [38]. This is common in systems with spin 1 or larger. Near such an anticrossing there is no first-order magnetic Zeeman shift. Consequently, spin flips of nearby host and active spins, which ordinarily introduce coherence by perturbing the local magnetic field of the qubit, no longer have a first-order effect. The complication is that the magnetic field is a vector so that the level anticrossing must exist in all three directions. Nonetheless such global level-crossings were found in Pr:YSO and were used to lengthen the qubit spin coherence lifetime by orders of magnitude. More important, the residual spin decoherence was found to decay as a quadratic exponential 2 in time, meaning it decays as e−(t/τ ) . This is critical because most quantum error correction schemes require the short-time decay to be slower than the usual linear exponential decay. Because this condition was satisfied in Pr:YSO, a version of bang-bang error correction was successfully applied to give the observed half-minute coherence times. Manipulation of spin qubits in SHB materials is generally done using RF coils similar to those used in liquid NMR. Recently, optical Raman transitions have been explored as an alternative to this, in which case the spin-qubits are manipulated by lasers instead of an RF coil. The advantages of this are


497

twofold. First, the gate time can be made faster because it depends on the optical Rabi frequency, rather than that of the spin transition. One reason for this is that spin transitions are generally magnetic dipole allowed transitions, whereas optical transitions are often electric dipole allowed. Another reason is that it is often easier to insert strong laser fields into a cryostat than strong RF fields. Second, the selectivity of qubit excitation can be improved considerably because only spins with the correct optical transition frequency and spin transition frequency are manipulated. Additional spatial selectivity exists because the optical laser beams can be focused down to microns, and only the illuminated part undergoes qubit manipulations. This is especially important for algorithms like those designed for a Type II quantum computer; see Section 14.5. The real power of optically addressed NMR lies in multiqubit manipulations. The optical “handle” allows several options to increase scalability. First, the relatively long range of optical interactions frees NMR from the restrictions imposed by near-neighbor interactions. An example of this is an ensemble-based two-qubit gate demonstration in Eu:YSO involving ions separated by ∼100 nanometers [79], which is orders of magnitude larger than distances required by conventional NMR. In this demonstration, a series of optical pulses refocuses a “target” optical qubit with a different phase depending on whether the “control” qubit is excited. Because these qubits are defined only by their transition frequency, neither the exact location nor number of spins located in between is unimportant. This demonstration also illustrates the interesting fact that optical transitions in some SHB materials have a coherence lifetime that is similar to that of many room temperature NMR transitions. In principle, long-range optical interactions such as in the Eu:YSO example are scalable. In practice, however, the Eu:YSO demonstration experiment is not very scalable because well-defined pairs of qubits must be distilled out of a random ensemble [94], and this incurs an exponential penalty with number of qubits. To make this technique more scalable one approach would be to apply it to special solid-state pseudo-molecules. These pseudo-molecules exist in a number of stoichiometric crystals, the most interesting of which are those containing europium, for example EuVO4 or Eu2 O3 because of their narrow optical transitions at low temperatures [48, 78]. In these pseudo-molecules, localized defects have a large effect on the optical transition frequency of the Eu ions. Up to 50 optical transitions can be easily resolved in these materials. Assuming that all the defects are identical, each optical transition would correspond to a Eu spin system in a well-defined location near the defect center, thereby producing a pseudo-molecule. By using the long-range optical coupling demonstrated in Eu:YSO, one could in principle construct up to a 50-qubit quantum computer without most of the usual scaling limitations of NMR. To achieve scalability beyond 100 qubits, single-spin manipulation is preferred. The excitation and especially detection of single spins in a solid is a

498


very active area of research. Much of this research is based on a proposal to build a quantum computer using qubits consisting of nuclear spins of a phosphorus atom implanted in a silicon host [59]; see Figure 14.12. 31 P nuclei in spin-free 28 Si have a spin population lifetime on the order of hours at ultracold temperatures, with coherence lifetimes on the scale of tens of milliseconds so far. Single-qubit manipulations would be done with the usual NMR pulse sequences. However, to avoid driving all the qubits at once, an off-resonant RF field is applied and the active qubit is tuned into resonance when desired by using the interaction with its electron spin. The electron spin in turn is controlled by distorting the P electron cloud with a voltage applied to a nearby gate, called the A-electrode. To achieve two-qubit logic, the electron clouds of two neighboring P atoms are overlapped using a J-electrode. The resulting exchange interaction between the two electrons can then be transferred to the P nuclei using RF and/or gate pulse sequences. Because the P atom has electron spin, Boltzmann initialization can be used, although a number of faster alternatives such as spin injection are being explored. For readout, the nuclear spin state is transferred to the electron spin which in turn is converted into a charge state via spin exchange interactions with a nearby readout atom. The charge state is then detected with a single-electron transistor. The Hamiltonian for P in Si is given by [59]: H = μB Bσze − gn μn Bσzn + Aσ e · σ n ,

(14.58)

where μB is the Bohr magneton, μn is the nuclear magneton, gn is the nuclear g-factor, B is the applied magnetic field (assumed parallel to z), and σ are the Pauli matrices, with superscripts e and n for electron and nuclear. For two coupled qubits the Hamiltonian becomes: H = H(B) + A1 σ 1n · σ 1e + A2 σ 2n · σ 2e + Jσ 1n · σ2e ,

(14.59)

where H(B) is the magnetic field part and the superscripts 1 and 2 refer to the two spins, µ ¶5/2 r e2 4J ∼ e(−2r/aB ) , (14.60) = 1.6 εaB aB where r is the distance between P atoms, ε is the dielectric constant of silicon, and aB is the effective Bohr radius. Many of the more challenging operations proposed for the P in Si quantum computer have recently been demonstrated in quantum wells/dots using individual electrons and/or excitons [102]. In most of these experiments electron spins, rather than nuclear spins, are the qubits. Unfortunately, these experiments are usually done in GaAs which is not a spin-free host and so decoherence is a problem. Spin-free semiconductor hosts exist, but fabrication in these systems is not yet as mature (except for silicon).


499

Fig. 14.12 Operation of the P doped Si quantum computer. (a) Single qubits are tuned into resonance with applied microwave field using voltages on A-electrodes to distort weakly bound electron cloud. (b) Two-qubit gates are enabled by using voltages on J-electrodes to overlap neighboring electron clouds, thereby switching on spin exchange coupling.

Although the exciting prospect of single electron and/or nuclear spin detection is currently a technical challenge for most solids, it was done long ago in nitrogen-vacancy (NV) diamond [47]. In this optically active SHB material, single-electron spin qubits are routinely initialized and read out with high fidelity at liquid helium temperature. In fact, the fidelity is so high that it begins to compare to trapped ions [52]. The electron spin coherence has also been transferred to nearby nuclear spins to perform (nonscalable) two-qubit logic [53]. Single-qubit logic is usually done with RF pulses, but optical Raman transitions between spin sublevels have been observed in NV diamond under certain experimental conditions [49]. Two-qubit gates can be performed using the electron spin coupling between adjacent qubits. Initialization of such a two-qubit system consisting of a NV and nearby N atom has recently been demonstrated using optical pumping. Scalability of this system can be achieved using an electron spin resonance (ESR) version of a Raman transition to transfer this spin coupling to the nuclear spins [113]. This ESR Raman has already been demonstrated for single NV spins. The spin Hamiltonian for NV diamond is given by [92] ← → 1 1 → H = D(Sz2 − S 2 ) + βe B← g e S + S A I + P (Iz2 − I 2 ), 3 3

(14.61)

← → where D is the zero-field splitting, B is the applied magnetic field, A is the electron—nuclear hyperfine coupling tensor, S and I are the electron and nuclear spin vectors, and P is the nuclear quadrupole contribution. With optical Raman, it should be possible to use long-range optical dipole— dipole coupling [80], or eventually cavity-based quantum electrodynamic

500


Fig. 14.13 (a) Two distant qubits coupled by vacuum mode of cavity using cavity QED. (b) Possible implementation of a multiqubit bus using a photon bandgap cavity in NV diamond. (c) Illustration of the use of optical Raman with cavity QED to couple spectrally adjacent qubits.

(QED) coupling, to perform two-qubit spin logic; see Figure 14.13. If successful, this will be highly scalable because the optical transition frequency can be tuned where desired using dc electric Stark shifts created by gate electrodes (A-electrodes). By requiring both optical laser frequency and electrode voltages to be correctly tuned, qubit excitation becomes very selective and two-qubit coupling only exists when needed, in contrast to the usual case in NMR. Electron spin population lifetimes up to minutes have so far been observed at low temperature, with coherence lifetimes up to 0.3 milliseconds at room temperature [52]. Interestingly, the optical initialization and readout still works at room temperature (although with less fidelity), as well as the ESR Raman transitions. This raises the intriguing question of whether a room-temperature solid-state NMR quantum computer can eventually be built using NV diamond.


501

14.4 Shor’s Algorithm and Its Experimental Realization Through the rest of the chapter, we describe two applications of the NMR quantum computer: Shor’s algorithm and a lattice algorithm. Entangled states are extremely important in quantum computation. Entanglement, together with superposition, gives a quantum computer the power to perform massively parallel computation and thus makes it particularly suitable for computing certain complex problems. Shor’s algorithm for the factorization of integers aimed at decryption is a special example of a “killer ap” of quantum computing [30, 96]. Recently, a successful experiment has shown the potential capability of the implementation of Shor’s algorithm, although it is still very simple and tentative. In [111], Vandersypen et al. factor 15 into 3 times 5. That work has shown the liquid NMR quantum computer to be the most successful quantum computer so far.

14.4.1 Shor’s Algorithm It is not difficult to factor a composite integer (i.e., nonprime) into prime numbers when that integer is small, but the computation burden grows rapidly when the number increases. The currently most efficient algorithm, the num1/3 2/3 ber field sieve, requires an exponential running time ec(log n) (log log n) , where n is the number to be factored and clearly log n is proportional to the number of the bits needed to store this number. This makes it practically impossible to factor a large number using a classical computer. This difficulty is used to construct several cryptosystems, such as the RSA public key cryptosystem [95]. Peter W. Shor has shown that this problem can be solved in polynomial running time instead of exponential time by using the quantum computer. A more accessible account of Shor’s algorithm is given by Lomonaco [75]. Let n be an odd integer to be factored, and choose another random integer x less than n. We require x to be coprime with n; otherwise, we find a factor of n immediately by the Euclidean method. It is then known that function f (s) = xs mod n is periodic. The period of f (and also of x) is the smallest integer r such that xr = 1 mod n. For example, when n = 15 and x = 3, the moduli of xs , with s being 1, 2, 3, . . . , are 3, 9, 12, 6, 3, 9, 12, 6, . . . , and the period is 4. Now we check the period r. If r is even, r = 2t, then x2t −1 = (xt +1)(xt − 1) = 0 mod n, so either xt −1 or xt +1 has a common factor with n. A classical computer can use the Euclidean algorithm to compute the greatest common divisors, denoted as gcd(xt +1, n) and gcd(xt −1, n), in polynomial time. It is possible that we only obtain the trivial factors 1 or n using the x we choose. This happens only when xt = −1 mod n, because xt − 1 = 0 mod n cannot happen with r being already the smallest integer such that xr = 1 mod n.

502


Fortunately it has been proved that the probability to meet such a bad x is at most 1/2k , where k is the number of distinct prime factors of n. Because k is at least 2, the probability is still large enough for us to find a good x, which has an even period r and xt 6= −1 mod n. A quantum computer can find the period r because of the speedup afforded by the quantum Fourier transform (QFT). Let us have two b-qubit registers. We select b large enough such that we can observe many periods. At the beginning, we set the two registers to state |0i. Then we randomize the first register to a new state S−1 1 X |ψ1 i = √ |ki|0i, (14.62) S k=0

where S = 2b , the number of the total b-qubit states of the first register, with b large enough such that 2n2 > S > n2 . We now design a certain series of pulses to compute f (k) = xk mod n, and change the quantum state to S−1 1 X √ |ψ2 i = |ki|f (k)i. S k=0

(14.63)

Now, we apply QFT [90] to the first register in (14.63), which is a unitary transform mapping every |ki to another state: S−1 1 X 2πiuk/S |ki → √ e |ui. S k=0

(14.64)

Then the quantum state of the system changes to |ψ3 i =

S−1 S−1 X 1 X |ui e2πiuk/S |f (k)i. S u=0

(14.65)

k=0

Assume that f (k) has period r, and we write k = d + jr such that 0 ≤ d < r, where d is the remainder of k after it is divided by r and j ranges from 0 to A, the largest integer such that Ar < S. This way, we can write |ψ3 i as S−1 r−1 A X X 1 X 2πiud/S |ui |f (d)ie e2πiurj/S I(d+rj<S) , |ψ3 i = S u=0 j=0 d=0

where I(d+rj<S) = 1 when d + rj < S, and 0 otherwise. If S = (A + 1)r, I(d+rj<S) = 1 for every d and j. If S 6= (A+1)r, it is still reasonable to ignore the difference and let I(d+rj<S) = 1 everywhere because we have chosen S large enough. In this case, we let


503

¶ µ A 1 X 2πiurj/S 1 1 − e2πiur(A+1)/S bu = e = ; S j=0 S 1 − e2πiur/S

(14.66)

thus our quantum state is now |ψ3 i =

S−1 r−1 1 XX bu e2πiud/S |ui|f (d)i. S u=0 d=0

We can now measure the first register, and we want to find such a u, for which there is an l satisfying ¯ ¯ ¯u ¯ ¯ − l¯≤ 1 . (14.67) ¯S r ¯ 2S

There are about r such us, and it has been estimated that the probability to find such a u is at least 0.4 [90]. Because 1/2S < 1/2n2 , and we know that r < n, there is at most one fraction k/r satisfying the condition and we can use continued fraction expansions to find the fraction. If k and r are coprime, we obtain r as the denominator of the fraction. If not, we only find a factor of r. If r is odd or xr/2 does not give a useful result, we choose another x and try again. It may be necessary to try several (of the order O(log log n)) times until r is successfully found, but the overall running time is still reasonable.

14.4.2 Circuit Design for Shor’s Algorithm Before we introduce the experiment by Vandersypen et al. [111], we extend the above discussion a little further to the case when r divides S. Now S/r becomes an integer and (14.66) always holds so that S doesn’t have to be very large. Moreover, (14.67) becomes an identity u=

l·S ; r

(14.68)

that is, r is the denominator of the fraction u/S after canceling the common factor between u and S if l and r are coprime. The integer 15 falls into this situation. The possible x can be 2, 4, 7, 8, 11, or 13. When we choose x to be 2, 7, 8, or 13, the period r is 4. In other cases, r is 2. The period r divides S = 2b in both cases. Only two qubits at most are required to compute one period of f . In the experiment, three qubits are used to obtain more periods. Vandersypen et al. used liquid NMR to realize Shor’s algorithm in factorizing 15. The sample in the experiment is a custom-synthesized material whose molecules have five 19 F and two 13 C, so it has seven qubits ready for use. Those seven qubits are divided into two registers, three to store the number

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k (the first register, represented by |k2 k1 k0 i) and four to store the modular exponentiation y (the second register, represented by |y3 y2 y1 y0 i); see Figures 14.14 and 14.15. The total Hamiltonian is H=

7 X 1 i=1

2

ωi σzi +

X

2πJij σzi σzj .

i<j

Each run of the experiment consists of four steps. In the first step, the sample is initialized to a certain pseudo-pure state; in the second step, a series of specially designed pulses is applied to realize the computation of modular exponentiation; in the third step, QFT is applied to the first register; finally, the period was obtained through the reading of the spectrum. The system begins from thermal equilibrium, where the density matrix is given by ρ0 = e−H/kT ≈ I − (H/kT ). A suitable initial pseudo-pure state |ψ1 i = |0000001i is obtained by the temporal averaging method. Although it is difficult to design a general circuit for the modular exponentiation, it is easy to “hard-wire” for this special case in consideration. As the exponent k can be written as k = k0 + 2k1 + 4k2 , we can change the modular exponential xk mod 15 into successive operations of modular multiplications i by x2 ki , with i = 0, 1, 2, applied to the second register y beginning from y = 1. When i = 0, y · x = x = 1 + (x − 1), so the multiplication is actually a controlled-addition with (x − 1) in the case k0 = 1. For x = 7 = (0111)2 , it is equal to flip the states of y1 and y2 (y = (0001)2 before the multiplication). For x = 11 = (1011)2 , the same reasoning shows that we only have to flip the states of y3 and y1 , depending on k0 . Gates A and B in Figures 14.14 and 14.15 accomplish the modular multiplication xk0 . The situation is a little more complicated for i = 1. We only discuss the situation when k1 = 1, because y will not change when k1 = 0. Different strategies are needed for x = 7 and x = 11. When x = 11, because 112 = 121 = 15 × 8 + 1, y × 112 = y (mod 15). We need to do nothing and the same result holds for the third qubit k2 . When x = 7, we can design the circuit by first investigating the following identity. y · 72 = y · 4 mod 15 = (y0 + 2y1 + 4y2 + 8y3 ) · 4 mod 15 = (4y0 + 8y1 + 16y2 + 32y3 ) mod 15 = (y2 + 2y3 + 4y0 + 8y1 ) mod 15 = (y2 · 20 + y3 · 21 + y0 · 22 + y1 · 23 ) mod 15. It shows that the modular multiplication can be achieved by exchanging the first qubit y0 with the third qubit y2 , and the second qubit y1 with the fourth qubit y3 . In Figure 14.14, gates C, D, and E are used to accomplish the former, and gates F, G, and H the latter. Further simplification of the


1 2 3 4 5 6 7

T e m p o r a l

a v e r a g i n g

H H H

505

H

Rπ/2

H

Rπ/4 Rπ/2

H

ABCDE FGH Fig. 14.14 The quantum circuit for the (hard) case for the realization of Shor’s algorithm (x = 7). From top to bottom, the qubits are k2 , k1 , k0 , y3 , y2 , y1 , and y0 , respectively, in sequential order.

1 2 3 4 5 6 7

T e m p o r a l

a v e r a g i n g

H

H H H

Rπ/2

H

Rπ/4 Rπ/2

H

AB Fig. 14.15 The quantum circuit for the (easy) case for realization of Shor’s algorithm (x = 11). From top to bottom, the qubits are k2 , k1 , k0 , y3 , y2 , y1 , and y0 , respectively, in sequential order.

circuit can be made. Because the control bit y3 is |0i before gate C, that gate can just be omitted. Gates H and E have no effect on the period; they can be omitted, too. The circuit design for the quantum Fourier transform is just a standard design; see, for example, [30, Figure 5]. It has three Hadamard gates and three controlled-phase gates. Figures 14.14 and 14.15 show the circuit designs for y = 7 and y = 11. About 300 pulses are used in the experiment in total and it takes about 700 ms to accomplish all steps in the case of x = 7.

14.4.3 Experimental Result Readout of the experiment needs a careful interpretation of the data. Because an NMR sample consists of many molecules, the readout is the average value of u from all molecules instead of the reading from a single molecule.

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Both qubits k0 and k1 are found to be in state |0i after the extraction of the spectra [111] for the easy case of x = 11, whereas qubit k2 is in an equal mixture state of |0i and |1i. Thus the possible u can be 0 and 4, that is, 000 and 100 in binary form. From (14.68), r can be obtained as r = 8/4 = 2, and the greatest common divisors are computed as gcd(112/2 + 1, 15) = 3 and gcd(112/2 − 1, 15) = 5. In the case of x = 7, the spectra in [111] indicate that only qubit k0 is in state |0i, and both qubits k1 and k2 are in an equal mixture of states |0i and |1i. Thus u is in a mixture of states |0i, |2i, |4i, and |6i. We can see that the period of u is 2, thus the period of the modular exponent r is 8/2 = 4. The factors of 15 can finally be obtained as gcd(74/2 − 1, 15) = 3 and gcd(74/2 + 1, 15) = 5.

14.5 Quantum Algorithm for Lattice-Gas Systems In the previous sections, we have explained how to construct a quantum computer using liquid NMR and illustrated a successful experiment. We have taken it for granted that the coherence can be maintained long enough and different qubits can be entangled even if they are separated far apart in space. Unfortunately, these assumptions are not always practical and in fact they constitute great obstacles to overcome. The problem becomes more serious when more qubits are involved. Type-II quantum computers are proposed to alleviate this problem. A Type-II quantum computer is composed of a network or array of small quantum computers interconnected by classical communication channels [120]. Instead of the global coherence and entanglement, only local coherence and entanglement within every small quantum computer, called a node, are required, and the difficulty faced by the centralized quantum computer is dramatically eased. The wave function of the whole Type-II quantum computer system is a tensor product over all the nodes: |ψ(t)i = |ψ(x1 , t)i ⊗

···

⊗ |ψ(xN , t)i,

(14.69)

where N is the number of nodes. The lattice-gas algorithm (LGA) is specially suited for this structure. Every computation cycle can be broken up into three steps with two intermediate states |ψ 0 i and |ψ 00 i: |ψ 0 i = Cˆ |ψ(t)i , |ψ 00 i = Γ |ψ 0 i , |ψ(t + 1)i = T |ψ 00 i ,

(14.70)

where Cˆ is a unitary operator acting locally on every node, Γ is a projection operator, such as a measurement, and T is the streaming operator that


507

exchanges information among nodes. The Type-II quantum computer takes advantage of parallelism in two ways: one classical, where all the nodes work simultaneously; the other quantum, where quantum entanglement is still kept inside every node. Because measurement is applied and the system is reset at the end of every computation cycle, the coherence only needs to be maintained for a short time.

14.5.1 Quantum Algorithm for a Lattice-Gas Model Consider a one-dimensional diffusion equation without boundary condition ∂ρ ∂2ρ , = ∂t ∂x2

(14.71)

where ρ is the mass density or temperature function along the x-axis. Using the finite difference method, we can write a finite difference approximation to solve the above partial differential equation numerically: ρ(x, t + τ ) − ρ(x, t) ρ(x + l, t) − 2ρ(x, t) + ρ(x − l, t) , = τ l2

(14.72)

where τ is the time step size and l is the space step size. From physics, we know that the above equation may be studied by a lattice-gas algorithm. Without loss of generality, we assume that τ and l are normalized so that the difference equation can be written as ρ(xi , k + 1) − ρ(xi , k) =

1 (ρ(xi+1 , k) − 2ρ(xi , k) + ρ(xi−1 , k)). 2

Points xi are evenly distributed along the x-axis, also called nodes. To study the above equation, two functions, f1 (xi , k) and f2 (xi , k), called channels, are defined for each node xi at time k. The set of values of these two functions is called the state of node xi . Any physical observable, such as the density function ρ(xi , k), is a function of the state at the node. The evolution of the lattice, or the state of all nodes, consists of two operations: collision and propagation. A collision is a local operator only defined by the state of the node itself. The propagation operator transfers information from one node to another and the state at one node changes according to the state of other nodes. This is completed by defining a velocity vector for every channel which gives the information flow a direction. In our special example here, information in the two channels flows in opposite directions. After propagation, one channel gets its new value from its left neighbor, and the other from its right neighbor. This LGA is completed with a Type-II quantum computer by J. Yepez of the Air Force Research Laboratory and M.A. Pravia et al. of the De-

508


partment of Nuclear Engineering at MIT [8, 93, 119, 120]. The actual result is not as good as desired, but improvement is still possible. To store a floating-point number, a classical computer uses a register with 32 or 64 bits, depending on the machine. Quantum computers presently have difficulty in doing it the same way as classical computers because there is not yet the technology for 32 or 64 qubits. In this quantum lattice-gas algorithm, a two-qubit system is proposed for every node. The two qubits are represented by |q1 (xi , k)i and |q2 (xi , k)i, respectively, and p p |q1 (xi , k)i = f1 (xi , k)|0i + 1 − f1 (xi , k)|1i, p p |q2 (xi , k)i = f2 (xi , k)|0i + 1 − f2 (xi , k)|1i.

The state of the whole system |ψ(xi , k)i at node xi and time k is a tensor product: |ψ(xi , k)i = |q1 (xi , k)i|q2 (xi , k)i p p = f1 (xi , k)f2 (xi , k)|00i + (1 − f1 (xi , k))f2 (xi , k)|10i p + f1 (xi , k)(1 − f2 (xi , k))|01i p + (1 − f1 (xi , k))(1 − f2 (xi , k))|11i.

Quantities f1 (xi , k) and f2 (xi , k) are the probabilities of occurrence of the state |0i for qubit 1 and 2, respectively, corresponding to the two channels, and 1 − f1,2 (xi , k) are the occurrence probabilities of the state |1i. Because the states are normalized, 0 ≤ f1,2 (xi , k) ≤ 1, and we let ρ(xi , k) = f1 (xi , k) + f2 (xi , k). It is noted that our Type-II quantum computer assigns ρ a continuous value (a function of the occurrence probabilities) rather than a discrete value as a digital computer does. An array of two-qubit systems is used in the computation, corresponding to a series of nodes. The quantum LGA here has three steps in every cycle that complete a step of the finite difference algorithm computation: collision, measurement, and reinitialization. The last two composed are equal to one propagation operation in a normal lattice-gas algorithm. Because the propagation needs information exchange among different nodes, measurement and classical communication are needed to accomplish one operation. We map the quantum state to a vector in C4 as that given in (14.29). In the collision step, a unitary operator is applied simultaneously to all nodes: |ψ(xi , k)i = U |ψ(xi , k)i, where

⎡

1

⎢0 ⎢ U =⎢ ⎣0 0

0 1 2 1 2

− + 0

0 i 2 i 2

1 2 1 2

+ − 0

0 i 2 i 2

⎤

0⎥ ⎥ ⎥. 0⎦

1

(14.73)


509

The new occurrence probabilities of the state |0i of the two qubits after the operation, f 1 and f 2 , can be computed using ⎤ ⎡ 1000 ⎢0 1 0 0⎥ ⎥ f 1 = hψ|n1 |ψi, n1 = ⎢ ⎣0 0 0 0⎦, 0000 ⎡ ⎤ (14.74) 1000 ⎢0 0 0 0⎥ ⎥ f 2 = hψ|n2 |ψi, n2 = ⎢ ⎣0 0 1 0⎦, 0000

leading to

f 1 (xi , k) = 12 (f1 (xi , k) + f2 (xi , k)), f 2 (xi , k) = 12 (f1 (xi , k) + f2 (xi , k)). The collision operator is actually doing a job of averaging. The state after the collision is also called the local equilibrium. In the second step, a measurement is applied at every node and f 1,2 (xi , k) of all the nodes are retrieved for future use. In the third step, using information from the measurement from the previous step, the states of all the nodes are reinitialized to a separable state q q |q1 (xi , k + 1)i = f 1 (xi+1 , k)|0i + 1 − f 1 (xi+1 , k)|1i, (14.75) q q |q2 (xi , k + 1)i = f 2 (xi−1 , k)|0i + 1 − f 2 (xi−1 , k)|1i.

It can be seen that the second and third steps have accomplished the propagation operation. At node xi , the new state of channel one is acquired from the same channel of its right neighbor node xi+1 , and channel two acquires its state from its left neighbor. It is complicated here only because the communication between two quantum systems is difficult. To see how this LGA works, let us begin from a local equilibrium state, f1 (xi , k) = f2 (xi , k) = ρ(xi , k)/2, where the states come from a collision operation (step 2). We list the f1,2 around position xi before the third step in two rows f1 :

···

f2 :

···

ρ(xi−2 ,k) 2 ρ(xi−2 ,k) 2

ρ(xi−1 ,k) 2 ρ(xi−1 ,k) 2

ρ(xi ,k) 2 ρ(xi ,k) 2

ρ(xi+1 ,k) 2 ρ(xi+1 ,k) 2

··· ···

and after the third step f1 :

···

f2 :

···

ρ(xi−1 ,k) 2 ρ(xi−3 ,k) 2

ρ(xi+1 ,k) ρ(xi+2 ,k) ρ(xi ,k) 2 2 2 ρ(xi−2 ,k) ρ(xi−1 ,k) ρ(xi ,k) 2 2 2

··· ··· .

510


We can see that the row of f1 (channel one) moves left and the row of f2 (channel two) moves right. According to our definition, the new value of ρ is the sum of f1 (xi , k + 1) and f2 (xi , k + 1); that is, ρ(xi , k + 1) = 12 (ρ(xi+1 , k) + ρ(xi−1 , k)). It is easy to check that ρ(xi , k + 1) − ρ(xi , k) =

1 (ρ(xi+1 , k) − 2ρ(xi , k) + ρ(xi−1 , k)), 2

as desired. Applications of the Type-II quantum computer with quantum LGA also have been reported in the simulation of the time-dependent evolution of a many-body quantum-mechanical system [121], solution of a one-dimensional magnetohydrodynamic turbulence [105], representation of solitons [104], and other equations.

14.5.2 Physical Realization and Result The experiment in Section 14.5.1 uses a two-qubit molecule, chloroform, whose structure is shown in Figure 14.6. The hydrogen and carbon nuclei serve as qubits 1 and 2, respectively. The actual results obtained from the experiment are compared with simulation results [93]. After 12 steps, the error becomes very large. Imperfection in the decoupling sequences is blamed and it is believed that the problem can be mitigated when the technology is improved in the future. Extreme requirement of high accuracy in the control pulse and readout is a disadvantage of this Type-II quantum computer, because it uses a continuous representation (the probability of occurrence) instead of a discrete one. Thus, it is more vulnerable to the inaccuracy in the NMR operation. Small errors in every step accumulate and finally become intolerable. Repeated measurement and reinitialization ease the requirement for coherence time, but place a high requirement on the fidelity at the same time.

14.6 Conclusion In this chapter, we present the basic technology used to construct a quantum computer with liquid-state NMR. The successful experiments for many algorithms have shown that liquid-state NMR is capable of simulating a quantum computer and forms a testbed for quantum algorithms. It is so far the only technology available to realize a seven-qubit algorithm in the laboratory. One reason for its success is the robustness of the spin system which only interacts with the external magnetic field, and it is possible to maintain the coherence for a long time (from seconds to hours). Besides, over the 60-year history


511

of NMR spectroscopy, analytic tools have been developed for the purpose of chemical and medical applications, and exact description and dedicated coherence control of the dynamics of the quantum spin system are now available to achieve high accuracy in the pulse design and application. In fact, the experimental techniques established in NMR, especially the coherence control technology, can be easily transferred to other quantum systems if they have a similar Hamiltonian, and the research in NMR is therefore helpful for the development of other more complicated and powerful quantum computers. Liquid NMR has played a pioneering role in quantum computer technology development. But its lack of scalability has constituted a severe obstacle to its future applicability. However, in Section 14.3 we show new technology of solid-state NMR that has the potential to overcome liquid NMR’s difficulties. For solid-state NMR, under low temperature, the relaxation times of spins are typically very long, and the coupling between qubits is strong so that the control can be fast and easy. The small ratio of the gate time and the decoherence time makes more gates available, and more complicated algorithms can be tested. The nuclei can be cooled down easily and the spin system is highly polarized. The signal is much stronger so that fewer nuclei are needed. Even without the help of the gradient field and silicon technology, as we have mentioned, a quantum computer with 30 to 40 qubits is envisioned with designed molecules similar to that of the liquid-state NMR computer except that the ensemble is in a solid crystal state. This is already a quantum system that reaches the limit a classical computer can simulate. Although it is still not scalable and not a standard quantum computer, these small- and medium-scale quantum computers will help in the building of a scalable and working quantum computer. Acknowledgment We wish to thank the reviewer for constructive comments that have improved the quality of this chapter.

Appendix The Homeomorphism from SU(2) to SO(3) We use the same notation as in Section 14.2.3. Recall that U is a complex matrix in SU(2), and we want to find a mapping R from SU(2) to the space of 3 × 3 real matrices, so that R(U ) and U represent the same physical operation. Let v be a Bloch vector corresponding to the old state |φi and v0 to the new one. Then the following identity must be satisfied, v0 = R(U )v,

(14.76)

512


where U ∈ SU(2). Using the definition of Bloch vector and density matrix, we obtain v0 · A = U |ψihψ|U † − I0 (14.77) = U v · AU † , where A = [Ix , Iy , Iz ] and we use the dot to denote the inner product between vectors. The product of Ii and σj satisfies Tr(Ii σj ) = δij ,

i, j ∈ {x, y, z},

where σj are the Pauli matrices. By multiplying both sides of (14.77) with σk and taking the trace of both sides, we obtain vk0 = Tr(v0 · Aσk )

= Tr(σk U Ii U † )vi ,

(14.78)

where vk0 and vi are, respectively, the kth and ith entry of the corresponding vectors. We apply the summation convention in this section, where summing over repeated indices is implied unless otherwise stated. Comparing the above result with equation vk0 = R(U )ki vi , we obtain the desirable matrix R(U ): R(U )ki = Tr(σk U Ii U † ).

(14.79)

Thus we have constructed a mapping from SU(2) to the set of 3×3 matrices. Let us now tentatively accept the fact that the target matrix is real which we prove later. We first show that R(U ) is in fact a proper rotation matrix, that is, R(U ) ∈ SO(3), by proving that it is isometric and preserves the threefold product (cf. (14.82)). Let v = [a b c]T ∈ R3 . First note that we can find its norm by computing the determinant of a special matrix: ¯ ¯ ¯ c a − bi ¯ 1 ¯ ¯ det(v · A) = 4 ¯ a + bi −c ¯ (14.80) = − 14 (c2 + a2 + b2 ) = − 14 kvk2 .

Together with (14.77), we see that the transformation is isometric:


513

kv0 k2 = −4 det(v0 · A)

= −4 det(U v · AU † ) = −4 det(v · A)

(14.81)

= kvk2 .

Direct computation also shows the preservation of the threefold product, as follows. Let vl , l = 1, 2, 3, be three vectors on the Bloch sphere, v0 l = R(u)vl . We have ¡ ¢ Tr (v1 · A)(v2 · A)(v3 · A) = Tr(vi1 vj2 vk3 Ii Ij Ik ) = 4i vi1 vj2 vk3

ijk

(14.82)

= 4i v1 · (v2 × v3 ). The above identity can be used to show the preservation of the threefold product: ¡ ¢ v0 1 · (v0 2 × v0 3 ) = 4i Tr (v0 1 · A)(v0 2 · A)(v0 3 · A) ¡ ¢ = 4i Tr (U v1 · AU † )(U v2 · AU † )(U v3 · AU † ) (14.83) ¡ ¢ = 4i Tr (v1 · A)(v2 · A)(v3 · A) = v1 · (v2 × v3 ), and we see that R(U ) ∈ SO(3). To prove that the mapping is surjective, we introduce a parameterization of SU(2) in some exponential form. For any U ∈ SU(2), we can find a θ ∈ [0, 2π) and a unit vector n such that U (θ, n) = e−i(θ/2)n·σ " # cos θ2 − in3 sin θ2 − sin θ2 (n2 + in1 ) = sin θ2 (n2 − in1 ) cos θ2 + in3 sin θ2

(14.84)

= cos θ2 I − i sin θ2 n · σ,

where σ = [σx , σy , σz ]. Every combination of θ and n also corresponds to a complex matrix in SU(2). By using (14.8) and the equality U (θ, n)−1 = U (θ, −n), the element of R(U ) can be computed as R(U )ij = Tr(σi U Ij U † ) ¡ ¢ = Tr σi (cos θ2 I − i sin θ2 n · σ)Ij (cos θ2 I + i sin θ2 n · σ) = Tr(σi Ij cos2

θ 2

+ iσi Ij n · σ sin θ2 cos θ2

− iσi n · σIj sin θ2 cos θ2 + σi n · σIj n · σ sin2 θ2 ). We divide the trace into four parts and compute them separately:

(14.85)

514


Tr(σi Ij ) = δij , Tr(σi Ij n · σ) = i ijk nk , Tr(σi n · σIj ) = i Tr(σi n · σIj n · σ) = Tr(σi σk Ij σl nk nl )

ikj nk ,

= 12 Tr ((δik I + i ikm σm )(δjl I + i jln σn )nk nl ) = δik δjl nk nl − ikm jlm nk nl = δij ni nj + (2 − δij )ni nj − δij = 2ni nj − δij .

(14.86)

(14.87)

Substituting (14.87) and (14.86) back into (14.85), we obtain R(U )ij = cos2 2θ δij + sin2 θ2 (2ni nj − δij )

− ijk nk sin θ2 cos θ2 + ikj nk sin θ2 cos θ2 = cos θ δij + (1 − cos θ)ni nj + sin θ ikj nk ,

(14.88)

showing that R(U )ij is real. We also claim that R(U ) is a rotation about the axis n with angle θ in the three-dimensional Euclidean space by comparing it with the standard formula of a rotation matrix. Because every matrix in SO(3) can be regarded as a rotation about a certain axis with a certain angle, that matrix is now shown to be an image of some U ∈ SU(2). Thus the mapping is surjective. Finally, we want to show that this mapping from SU(2) to SO(3) is a homomorphism and to investigate the multiplicity. Let U and T be two matrices in SU(2), v be an arbitrary vector on the Bloch sphere, and v0 = R(U )v, v00 = R(T )v0 = R(T )R(U )v. Using (14.77) repeatedly, we obtain v00 · A = T (v0 · A)T † = T U (v · A)U † T † = R(T U )v · A, thus v00 = R(T U )v. Because v is arbitrary, R(V U ) = R(V )R(U ), implying that the mapping is a homomorphism. For the multiplicity of this mapping, we investigate the kernel of R, ker(R), which is an invariant subgroup of SU(2), and the quotient group SU(2)/ker(R) will then be isomorphic to SO(3). Suppose that U ∈ SU(2) and R(U ) = I3 , the identity matrix in SO(3). Then for any v on the Bloch sphere or R3 , we have equality v · A = U (v · A)U † , because v = Iv. Multiplying both sides with U from the right and using (14.84), we have θ θ θ θ v · A(cos I − i sin n · σ) = (cos I − i sin n · σ)v · A. 2 2 2 2 Subtract from both sides the term v · A cos (θ/2) I, (v · A) (n · σ) sin

θ θ = (n · σ) (v · A) sin . 2 2

Assuming that sin (θ/2) 6= 0, we can divide this factor from both sides to obtain (v · A) (n · σ) = (n · A) (v · σ).


515

Let v = [1, 0, 0]T . Using properties of Pauli matrices, we obtain n2 Iz − n3 Iy = 0. The above is possible only when n2 = n3 = 0. Trying other different v leads to n1 = 0, too. This is a contradiction, because we know that n is a unit vector. Thus we need sin (θ/2) = 0. In this case U = I or U = −I and it is easy to verify that these two are really mapped to the identity in SO(3). Now, we can conclude that the mapping we defined by (14.79) is a twoto-one homomorphism from SU(2) to SO(3) with kernel ker(R) = {I, −I}. The mapping is also surjective, so it defines an isomorphism from the quotient group SU(2)/ker(R) to SO(3). The two elements in the kernel, ±I, are in fact the same transformation for quantum systems because only the relative phase matters for a quantum system. For any O ∈ SO(3), the two elements in R−1 (O), U and −U for some U ∈ SU(2), represent the same transformation, too. Thus, nothing is lost if we employ SO(3) to represent the transformations of a one-spin quantum system.

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